ATWO-ASSET PORTFOLIO WITH TWO-FACTOR MODELS
The core of the factor models is the variance-covariance matrix of the risk factors, E, and the factor loadings of each factor, plus the constants and the residuals. The variance-covariance matrix of factors is common to all assets. The residuals of each model have a mean of zero and a variance that depends on the model and the risk decomposition varies accordingly with each asset. Residuals, for each asset, have no correlation with the factors, but they might have cross-correlations between themselves differing from zero. Accordingly, we make explicit the variance-covariance matrix of residuals. In this sub-section, we show that the portfolio can be dealt with as a single asset by using asset weights within the portfolio.
The Two-factor Model Applied to a Two-asset Portfolio
As above, we use double subscripts for variances and covariances of factors. The a.., where i =j, are the variances a2 of factor return X., and the a, for i ^ j, in this case a single value, o12, represents the covariance of factor returns Xt, where Stakes the values 1 and 2. The variance-covariance matrix of factors E is:
Note that in the above matrix, the double index refers to factors only, unlike the double index used in the two-factor model, where the first subscript refers to the factor and the second one to the asset.
We have two models, one for each of the two assets. There are two sets of sensitivities with same factors, plus residuals attached to each of the two assets. The factor models for each asset return are:
The coefficients of the two models have double subscripts, one for the asset and another for the factor. The first subscript refers to the factor and the second subscript refers to the asset. Note that the double subscripts are different for the variance-covariance matrix of factors, where they refer to factors only, and for the coefficients of the models, where they refer to factor (first subscript) and to asset (second subscript).
The constant of each model, are, respectively, RQ1 = 0.1 and RQ2 = 0.05. They do not contribute to variances and covariances, but they contribute to the expected return of the portfolio. The matrix of coefficients of the two models is as follows, ignoring the two constants because they do not contribute to risk.
We first derive the portfolio return and show that it has the same generic format as the single-asset two-factor model, although we need to input the weights of each asset in the formulas. The rest follows as in the single-asset model: variance-covariance matrix of asset returns and risk decomposition.