# Some definitions in the BARRA Predicted Beta model

I'm studying the BARRA Predicted Beta model, and the common factor covariance between portfolio $$p$$ and the return on the market $$m$$ is defined as the product of the transposed vector of the factor exposures for the portfolio, the factor covariance matrix, and the vector of the factor exposures for the market:

$$COV(r_p, r_m) = X_p^TFX_m$$

and the specific covariance is:

$$COV(r_p, r_m) = \sum_{i=1}^{N}{h_{pi}h_{mi}\sigma_i^2}$$

where:

$$F_{jk}$$ is the covariance between factors $$k$$ and $$j$$

$$\sigma_i^2$$ is the specific variance of asset $$i$$

$$X_{mj}$$ is the market's exposure to factor $$j$$

$$h_{pi}$$ is the holding of the portfolio in asset $$i$$

$$h_{mi}$$ is the holding of the market in asset $$i$$

I find those three definitions a little ambiguous. What is exactly meant by "holding"? Should it mean weights or weighted returns? What is the market's exposure to an asset?

Thank you.

• Hi: holding, $h_{pi}$ just means weight of asset in the portfolio which is total cap of asset in portfolio divided by total cap of portfoio. $h_{mi}$ is weight of asset in whatever index ( market portfolio ) one is referring to. ( so total cap of asset divided by total cap of index ). Feb 11 '19 at 4:39