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RISK FOR A SINGLE ASSET AND TWO-FACTOR MODELS

The risk calculations for a single asset have no other objective that providing the formulas, which, later, are applied to a portfolio by plugging-in the weights of each asset within a portfolio.

Assume that each single asset depends now on two risk factors. With a single asset, the sensitivities to each factor are 0.5 and 1. We derive the systematic and specific risks from the model for the single asset. The model for the single asset is:

The variance-covariance matrix E for factor returns is given in Table 32.4. The risk does not depend on the constant P0. There is no subscript for the asset since we consider a single asset. We use subscripts, with values 1 or 2, for the factors. The variances O 2 of factor returns X. are o... where i — /', and the a... for i ^ /', are the covariances of factor returns X. and x.

The analytical formula of the systematic variance V (Y), when we ignore the variance of the residual term, is, for the single asset, with factor coefficients Rj and P2:

Replacing Rj and Pj by their respective values, 0.5 and 1 :

The correlation coefficient between the two factors is given by po(Xj)o(X2) = cov{Xv X2).

The same result would be obtained from the matrix formula (Table 32.5), where the bottom right-hand cell contains the general variance, already calculated above.

TABLE 32.4 Variance-covariance matrix with a two-factor model

Variance-covariance matrix with a two-factor model

TABLE 32.5 Matrix formula for general variance of a single asset and a two-factor model

Matrix formula for general variance of a single asset and a two-factor model

In this matrix formula, we multiply the (1x2) transposed vector of coefficients pT by the square (2 x 2) variance-covariance matrix of factors E, and we multiply the row vector obtained by the (2 x 1) vector P of coefficients. The matrix formula for systematic variance is:

The specific risk in the case of a single asset is the variance of the single residual and the total risk, measured by the variance of the asset return, sums up the systematic and the specific risks:

 
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