Deriving The Variance of Equally-Weighted n-asset Portfolio
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 7th, 2010 at 8:03 am
Nice work! That’s my next reading, so this comes in on a perfect timing.
April 7th, 2010 at 9:31 pm
Thanks! How’s your progress Knarkh? Ready to tackle on level 2 in another 59 days?