Portfolio of Many risky Assets

There is a cumbersome formula for calculation the risk of portfolio that comprises three or more assets.

Is it right to assume that For a portfolio, that have “a”, “b”, and “c” assets, the variance will be equal to

Wa2σa2 + Wb2σb2 + Wc2σc2+ 2WaWbρa,bσaσb+ 2WaWcρa,cσaσc + 2WbWcρb,cσbσc

What are the chances that on the level 1, we are tested for the risk of portfolio with three or more assets?


Looks like you included all the necessary terms, at a quick glance it looks correct. Assuming of course your P terms denote the correct correlations between the different asset combinations.

I’d bet it’s likely they ask for the variance or standard deviation of a two asset portfolio, don’t think nearly as likely for three assets.

S2000 probably can give a much better answer of what’s likely or not to appear though, I’m sure he’s around at this time :slight_smile:


In a practice class of finance yeah, you will run matrixes with “n” assets, portfolio optimization, etc

On the CFA exams you will be tested on specific knowledge instead:

  1. Theory

  2. Simple calculations

One of my live professors stressed the “ugly formula” rule. If there is a big nasty formula that you can not intuitively create yourself (like FCFF in level 2), there’s a very good chance it will not be tested.

I’ll say that across practice tests from CFAI in level 1,2, and 3, I have seen the variance of a 2 asset portfolio pop up, but never for a portfolio of three or more assets.

But in the way you wrote this formula is pretty easy to memorize or make it more logical

2 assets portfolio consits of 2 squares and 2* weights stndev and correl

3 assets you have 3 sets of formula parts: 3 squares and addition of 3 parts of 2* weights stdev and correl of each two assets

Better is to memorize the fact that corr=cov/(stn devA *standdevB) so if you are given just cov between the assets skip standdev

Highly unlikley to have 3 assets on the day

It’s just as important to realize that increasing the number of assets increases the number of standard deviation terms by ONE, but increases the number of covariance terms by however many assets are already in the portfolio.

Therefore, when adding assets to a good sized portfolio, what matters most (as far as how it contributes to portfolio risk) is not the new assets own risk (i.e. its variance of returns), but rather its average covariance with all the other assets (i.e. its covariance with the portfolio’s returns).

Let’s try this again.