Risk Budgeting - Risk Contribution of Assets to the Portfolio

[emilioalzamora1]

Dear All, Dear @David Harper CFA FRM,

first, apologies if this turns out to be a quick fix/straightfoward problem but I do need some help/input with regard to the following:

This is from the CAIA (level 2) and it's about a 3-asset (classes) example of risk budgeting.

You are given the means of the 3 assets and their respective weights as well as variance-covariance matrix.

Asset (1): mu = 10% (weight 40%), variance = 0.0078
Asset (2): mu = 12% (weight 35%), variance = 0.01020
Asset (3): mu = 4% (weight 25%), variance = 0.04025

covariance (1,2) = 0.00746
covariance (1,3) = 0.00064
covariance (2,3) = -0.00372

Based on this we can simply calculate the variance of the 3-asset portfolio having:

sigma^2 (portfolio) = (0.4^2 * 0.0078) + (0.35^2 * 0.01020) + (0.25^2 * 0.04025) + (2*0.4*0.35*0.00746) + (2*0.4*0.25*0.00064) + (2*0.35*0.25*-0.00372) = 0.006578925 >>>> sigma (portfolio) = 0.08111 >>>> 8.11%

The question troubling me now is the fact how the contribution of each asset is calculated.
The CAIA uses the following formula for the risk contribution of each asset:

∂sigma(p)/∂ w(asset 1) * weight(asset 1) = cov(asset 1, portfolio)/sigma(p) * weight(asset 1) = ρ(asset 1, portfolio) * sigma (asset 1) * weight(asset 1)

This would imply that need to compute the covariance of asset 1 with the portfolio. But, how can this be done without having the correlation coefficient?

It says that the contribution of Asset 1 = ρ(asset 1, portfolio) * sigma (asset 1) * weight(asset 1)

where the value for ρ(asset 1, portfolio) * sigma (asset 1) is 7.26%.

In other words, we need to find the correl coeffcient between asset 1 and the portfolio to get 7.26%.

The total risk contribution amounts then: 7.26% * 40%

Any input is highly appreciated!

[The theory behind this is discussed in much more detail in Fabozzi 'Robust Portfolio Optimization and Management']

 

[David Harper CFA FRM]

Hi @emilioalzamora1 Yes, that'a a fun problem; thanks for the book pointer, I'm always looking for sources of Q&A inspiration, is that from the CAIA official text?

I wrote a similar question here https://forum.bionicturtle.com/thre...marginal-value-at-risk-var-calculations.7710/

... so i input your assumptions into a not-quite-cleaned-up version of the underlying XLS (see below), which is here at http://trtl.bz/0814-risk-contribution

You can see that i do get the same answer; mine takes correlations as inputs, where I used cov(1,2)/[σ(1)*σ(s)]. The stumbling block is always to your point about COV(Asset 1, Portfolio). But that's just the following please forgive my lazy syntax that is using '1', '2' and '3' as random variables which would otherwise be confusing: COV(1,P) = COV(1, w1*1 + w2*2, w3*3) or, if you like, COV(100*1, w1*1 + w2*2, w3*3). But COV(.) distributes so this is COV(1, w1*1) + COV(1, w2*2) + COV(1, w3*3) = w1*σ^2(1) + w2*COV(1,2) + w3*COV(1,3); those are in the 2nd row below under "COV(., P)." I hope that's helpful!

 

[emilioalzamora1]

Hi @David Harper CFA FRM,

you are simply a genius! I already said this but I do repeat it again. I very much appreciate your support even if it is not a question 100% related to the FRM material or your practice questions.

Yes, it is an original CAIA source: CAIA Level II (3rd edition) in the Chapter "Tactical Asset Allocation" (page 52-3). Only final results are given for illustrative purposes. If you don't have the book I am happy to send over a soft copy.

According to the lengthy derivation, such a question will definitely not show up on exam day (just because it's too time-consuming) - the level of difficulty is fine I suppose.

I was hoping to find something similar in Jorion but he has a different approach to Risk Budgeting as we do know from the second part of the FRM.