# Weighted average standard deviation

We all memorized that crazy formula for the standard deviation of a two-asset portfolio since Level I, right?

sqrt(w1^2 * std1^2 + w2^2 * std2^2 + 2 * w1 * w2 * std1 * std2 * corr1,2)

So, why is that not used when combining two corner portfolios? See Schweser Practice Test Volume 2 Exam 2 Morning, Question 6B, or just the example in Book 2 Page 182. It says using a simple weighted average is an “approximation” but it gives a totally different answer than using the full formula and assuming no correlation.

You know that the securities that compose adjacent corner portfolios are nearly identical; i.e., the portfolios probably differ by at most one or two securities. Thus, you would expect that adjacent corner portfolios would have quite high correlations of returns. The weighted average assumes a correlation of returns of +1.0, probably not far from the truth.

It is… You’re just assuming that they’re 100% correlated, since it’s easier to do it that way. It is impossible to use that formula unless you have correlation of each asset and using matrixs if there are 3 or more asset. The way they do it in level just assumes that correlation is 1. This over estimates correlation.

Try it out if you must. 6B exam two Schweser mock:

sqrt( 0.764^2*.058^2+.236^2*.078^2+2*1*0.764*.058*.236*.078 )= 6.27

.764*.058+.236*.078 = 6.27

^^^^^ Damn it S2000 you’re quick

0 – 60 in about 7 sec., cornering at 0.9g, top speed 150+ mph, and in Road & Track’s comparison test against a Porsche Boxster, beat it handily on the track.

Honda knows how to build 'em.

Awesome - thank you so much!

Actually you are assuming that they are 100% not correlated…and this is just a simplifying assumption since th purpose of this LOS concerns itself with a different angle on the topic and limits the data you have to bring to bear on the problem as per noted above. If anything using the exact calculation would enhance the sharp ratio since variability (risk would be reduced)

assuming there some degree of correlation between those returns

Actually your assuming it’s 100% correlated using the CFA’s formula. Which mean your over estimating your standard deviation, which in turn lowers ur sharpe making it more “prudent”.

sqrt(w1^2 * std1^2 + w2^2 * std2^2 + 2 * w1 * w2 * std1 * std2 (if we assume it’s 1) > sqrt(w1^2 * std1^2 + w2^2 * std2^2 (if we assume it’s 0). Anything lower 1 even with positive correlation would reduce risk.

The quickest way to test it is:

sqrt( w1^2 * std1^2 + w2^2 * std2^2 + 2 * w1 * w2 * std1 * std2 * 1)= w1*std1 + w2 *std2

They’re equal regardless of weights or standard deviation.

Also:

w1*std1 + w2*std2 =/= sqrt(w1^2*std1^2+ w2^2*std2^2). It only works with multiplication since sqrt is based on multiplication and not summation.

w1*std1 * w2*std2 = sqrt(w1^2*std1^2 * w2^2*std2^2)

Actually, you’re not.

Using the weighted average of the standard deviations implies a correlation of +1.0.

Honda is bring back the S2000 in 2016

Amen!

(I hope that it’s back to the 9,000 rpm red line.)