Calculating Standard Deviation of a portfolio?

CFA Level III
#1

Lets say you have a portfolio made up of Funds A & B and standard deviation and returns and allocation are given for both funds.

When are you able to use market value weights to calculated the the standard deviation of the portfolio vs using the long variance calculation (I forget the name of the formula)?

I don’t know when or why you would use one or the other? Anyone have any clarity?

thanks,

#2

Without the correlation of returns, you’re stymied. With it, you need to use the long formula. There’s no shortcut.

#3

AH! of course. not noticing these little things is gonna be the death of me.

#4

if you were using corner portfolios, just using the weighting of portfolios, no need to consider the correlations.

#5

Only because of the (usually incorrect) assumption that the correlation of returns of adjacent corner portfolios is zero one.

#6

Isn’t the assumption that the lineraity of risk and return also an incorrect assumption in using an artithmetic mean?

#7

I thought it was assumed the correlation between the corner portfolios was 1 rather than 0. That results in a slight over estimation of the portfolio standard deviation (as in reality the portfolios are most likely not perfectly positively correlated), which is seen as better than an underestimation and is therefore a valid approximation.

But yeah…I’m pretty sure the assumption is a correlation of 1, not zero.

#8

The concavity of the corner portfolio curves points to a lower standard deviation than calculated with a weighted average process, however, the weighted average process assumes no correlation.

#9

Well in Schweser it shows the calculation of a weighted average standard deviation for a combination of 2 corner portfolios:

Wa = 0.8 Wb = 0.2

Stdev of combination = 0.8(Stdev A) + 0.2(Stdev B)

and then states:

“professor’s Note: Mathematically this is the same result as if we used the more complex portfolio variance fomula but with all correlations of 1.0 - in other words, we allowed for no diversification between the corners. The corners themselves do consider diversification and correlation, making the final results quite accurate.”

So this seems to clearly state that the assumption is a correlation of 1.0 between the corner portfolios - hence the over estimation of StDev.

Or am I completely missing something?

#10

Edit: Seeing the math below, you’re right.

#11

Correl (A,B) = 1

Normal formula

VAR = w1^2*s1^2 + w2^2*s2^2 + 2*w1*w2*correl(A,B) * s1 * s2

When correl (A,B) = 1

formula reduces to

VAR = w1^2*s1^2 + w2^2*s2^2 + 2*w1*w2*1 * s1 * s2

= (w1*s1 + w2*s2)^2

So Std Dev = w1 s1 + w2 s2

#12

Absolutely correct.

My mistake. Don’t know what I was thinking.

(The “usually incorrect” still stands, however. This approximation generally overestimates the risk.)