 # Contribution to Portfolio Variance

So looking at the CFAI material (pg. 475-478 in Volume 4 for 2019) as well as the Kaplan Schweser material (pg. 70-74 in Book 4) and I’m trying to make sense of this new material added…

I believe I understand the Contribution of Asset A to Absolute Portfolio Variance (CVa) to be CVa = Wa x Cov (a, p), where we first find the Covariance of Returns between Asset A and the portfolio by calculating, for example, (Wa x Cova,a) + (Wb x Cova,b) + (Wc x Cova,c) and then to find A’s contribution to Total Portfolio Variance we just multiply Wa x the summation of what we just found.

Can someone explain Contribution of Factor A to Absolute Portfolio Variance and Contribution of Asset A to Relative portfolio variance in a detailed, step-by-step process?

I believe that Contribution of Factor A to Absolute Portfolio Variance is determined by:

1. Finding Variance attributed by A = [(βa x βa x Cov(a,a)] + [(βa x βb x Cov(a,b)]

2. Finding Contribution of Factor A to Absolute Portfolio Variance by dividing Step 1 by Total Portfolio Variance (assuming they give it to you or they give you portfolio standard deviation in which you square that).

Still confused on the steps for Contribution of Asset a to Relative Portfolio Value and cannot stress enough how both Kaplan and CFAI do a terrible job of explaining this…

I am still confused by this formula. Why do we use equation 8a to calculate contribution of asset towards the variance of portfolio? I thought equation 8a is for computation of portfolio variance and equation 8b is to compute contribution of variance by an asset towards total variance of a portfolio? Thanks!!

Anyone can help? It’s the equation on contribution of asset towards portfolio variance where

Equation 8a: calculation of total portfolio variance, V = ΣΣ (wi) (wj) cov (i,j)

Equation 8b: contribution of each asset to portfolio variance, CV = Σ (wi) cov (i,p)

I am unsure when to use which because according to cfai’s definition: the contribution of an asset to total portfolio variance is equal to the product of the weight of the asset and its covariance with the entire portfolio but in the example following the equation, they calculate the asset’s contribution to total portfolio variance using equation 8a instead of 8b. I find that the application and definition contradicting and confuses me.

Will appreciate if someone can chip in!

Anyone?

Total portfolio variance can be calculated w/ equation 8a above, and the contribution for, say Asset A, can be calculated by equation 8b. If they give the total portfolio variance, you only need to use 8b then divide by total variance. Otherwise, equation 8a is just repeating equation 8b for Assets A, B, C, etc. and summing them.

Relative (active) risk is appropriate when manager is concerned with performance relative to market index. Contribution of asset i to portfolio active variance (CAVi) is:

CAVi = (wpi - wbi)RCovip RCov= relative covariance of active returns between i asset i and the portfolio .

Assets contribute more if there’s a higher active weight and active returns are related to overall portfolio active returns. There was a question, not sure which test, stated the covariance as high or low to the portfolio, you just had to pick with asset was more accretive to the portfolio (the one with higher covariance.

Absolute Risk from Factors, where the contribution of factor i to portfolio variance (CVi) is: CVi=Betai x Covip

There could be a residual error, the active return not explained by the factors. these are the ost likely way they will be presented, this is ew equity material who knows, but in nearly all cases, when it comes to quant, understand the relationships. (usually the answer involves just a couple of steps, multiplying a x b. a is beta and c is cov. pick the factor with the highest CV.