Woah! What a mouthful title.
In any case, I’m back after such a long, long hiatus. There’s a rather long story behind that, but for some other time perhaps. Today I just want to derive the formula for the variance of an n-asset portfolio when it is equally weighted (i.e.: each asset has a weight of 1/w). In CFA Level II 2010 volume 6, page 392, the formula is stated to be this:

Yes... but how?
But how did we get there? The derivation is relegated into a footnote in the book, but we can derive this ourselves. Let’s do this, won’t take more than 10 minutes, I promise.
What is the formula for the variance of a portfolio with 2 assets? It’s this:

Covariance of a portfolio with 2 assets
This can be generalized into (where
):

Covariance of a portfolio with n assets, i <> j
So for portfolios with n assets, there will be n terms in the form of
, and
terms in the form of
. The
is our old friend from level 1 permutation. Specifically, the number of ways of obtaining an ordered subset of 2 elements (covariance between 2 assets) from a set of n elements (n assets).
Since this permutation always takes the form of
, we can simplify it to be
.
Let’s confirm this with the covariance formula for a portfolio of 3 assets (that is, n = 3):

We can see that there are 3
terms, and there are 6 (that is,
) terms in the form of
.
For a portfolio of 4 assets, there will be 4 and
terms, respectively.
Now, remember that our portfolio is equally weighted–meaning the weight of each asset is 1/n. So the formula becomes:

Substitute each asset's weight with (1/n), each asset's variance with average variance, and each covariance with average covariance
Which by simple multiplication becomes our original formula:

Q.E.D. Now off to Derivatives!
April 6th, 2010 | Category: CFA, Level 2 | Comments (2)