 # Calculating Standard Deviation of Portfolio

Schweser Book 2, Page 157, in the gray box: To calculate the portfolio’s standard deviation, it was a simple weighted average calculation, the assumption was the corner portfolios offer no additional diversification benefits. On the following page (Pg158), professor notes explaines that we are assuming a linear relationship, this is the upper limit to the standard deviation. OK! Schweser Book 3, Page 151, in gray box: To calculate the portfolio’s active risk (I understand it as the standard deviation of alpha, please correct me if I am not thinking this correctly), assuming no correlation between the satellite portfolios, it was using the full calculation approach(Squaring the weight, squaring the standard deviation, adding them all togheter, then take the square root) My question is these two examples seem to be identical. Why can’t I use the weighted average approach to calcuate the expected active risk for the satellite portfolio? What are the difference between these two examples? Thanks!

First is a linear approximation of the second.

I know that. Are both method acceptable?

I don’t know exactly what you are referring to here, but adding standard deviations just isn’t a good idea anywhere I can think of. I guess I agree with CSK’s comment but it’s an odd place to be doing a linear approximation (since a quadratic “approximation” is exact and it’s not like it’s some impossible function).

Ws, I had the same problem and this is how I think their logic was: 1 - In the case of Core-Satellite, the assumption is that the active return by the managers are uncorrelated ( correlation = 0), there is diversification effect. That’s why we use the full second order equation with corr = 0. 2 - In the case of the corner portfolios, it is assumed that we neglect any diversification effects. (That’s not the same as correlation = 0). That’s why we use the linear approximation in this case. Again, this is just my logic.

Mo34, I think the key is: if corr=0, does it imply any diversification benefits? Or if there is no diverification benefit, does it imply corr=0. What you said does make sense!

Corr = 0 , means diversification benefits, that’s correct. No diversification benefit means corr = 1. This is how to do it: SP^2 = w1^2 S1^2 + w2^2 S2^2 + 2 W1 W2 S1 S2 ( if correlation = 1) this can be re-written as: SP^2 = (w1 S1 + w2 S2) ^2 meaning SP = w1 S1 + W2 S2 et voila ! … No approximation needed … just correlation = 1

Cool, thanks Mo34

Huh? Where did we start discussing r = 1? r = 1 essentially means we just have the same investment perhaps levered differently.

Joey, The text indicates that we have to assume no diversification benefits when combining two corner portfolios on the efficient frontier to get a third one in between. I guess it’s because the two portfolios essentially contain the same securities but weighted slightly different, which means they will have a correlation close to 1. The other topic is constructing a portfolio with a core of index/enhanced indexing fund and some “satellite” holdings that are actively managed. Here the assumption is the returns of the actively managed funds are uncorrelated (r=0) I guess the logic here is that it would be stupid to pick up managers essentially doing the same thing.

mo34 Wrote: ------------------------------------------------------- > I guess the > logic here is that it would be stupid to pick up > managers essentially doing the same thing. Stupid me, that makes perfect sense now. R=0!!!

Oh…I see what’s going on here: In the first case, they are giving you a bunch of corner portfolios and then asking you to draw the efficient frontier. So you plot the corner points in a return/sd plane and then connect the dots. The idea is that all efficient portfolios lie on a line between the two corner portfolios so we can almost draw a line to get the frontier. Then the professor says something about this being a bound and not the actual frontier. As Mo points out, if r = 1 there is no diversification benefit and the sd’s are additive. So that means if you put sd on the x axis and return on the y axis, the lines you draw will always be within the efficient frontier. This has little to do with calculating the sd of a portfolio, but is just getting an inner bound on the efficient frontier which is “tight” at the corner points.

Joey, Sorry about not naming this post correctly!! “Calculating(estimate) Standard Deviation of a portfolio made up with corner portfolios.”

Which study session is this in? I vaguely remember reading about corner portfolios when skimming earlier, and I’ve already forgotten what they are. (but haven’t studied that section in earnest yet).

Corner portfolios are mentioned in SS7(asset allocation). Core-satellite portfolios are mentioned in SS10(equity managment)

Thanks, I remember skimming through asset allocation. Corner portfolios sounded both interesting and made sense to me then… I’ve just since forgotten. :-o! Oh dear… it’s starting early this year!