The Two-asset Portfolio Return
The risk of the portfolio depends on weights applied to each asset. The weights are indexed by the asset index "/" and sum up to 1. Weights are the ratios of initial value of each asset to portfolio value, and they are constant, and equal to their initial values. The portfolio return, Fp, is the weighted average of each asset return.
The portfolio return is now that of a sum of weighted asset returns, weights being constant.
The portfolio return is a linear model of the two factors. The coefficients are the weighted values of the sensitivities of the two assets. The constant is the weighted value of the constants of the two assets. The residual is the sum of weighted values of the two standalone (single-asset) residuals. This shows that we can proceed as if we had one standalone single asset with the constant, the coefficients of factors, the constant and the residual being the weighted values of the single asset model for the coefficients, constant and residual. The portfolio return is explained by a linear two-factor model.
For calculating the coefficients of this set of two models of each asset return, we need the set of weights and the constants. The weights are 0.7 and 0.3 for assets 1 and 2, or, in vector notation wT — (0.7, 0.3)T. The constants are 0.1 and 0.05, for assets 1 and 2, or (0.1, 0.05).
The weighted constant is calculated as above: 0.7 x 0.1 + 0.3 x 0.05 — 0.085. It can also be written as a vector product. The (1x2) row vector of constants P0T multiplied by the column (2x1) vector of weights w is the constant of the portfolio return model, or P0Tw:
The column (2x1) vector of weighted coefficients of factors X and X2 is obtained as above as weighted averages of factor coefficients. They are:
They can also be calculated as the product of the square matrix of coefficients by the column vector of weights, or pw:
Similarly, the linear function of residuals is the product of the row (1x2) vector of residuals by the column (2x1) vector of weights, or £Tw:
The global model for the weighted portfolio return, or Fp, is:
Fp = 0.085 +0.680^ + 1.080X2 + 0.7^ + 0.3e2
We end up with a single linear function of the same risk factors that relate to each asset return. But we added a linear function of the residuals of each model. The set of two-factor models, one for each asset, simplifies to a single two-factor model of the portfolio return, with two weighted coefficients, a single weighted constant, and a single weighted residual.
The expected portfolio return is the sum of expectations, whatever the covariances, except that we need to introduce the asset weights within the portfolio. The equation representing Fp as a two-factor model has weighted constants, weighted coefficients and weighted residuals. Applying the expectation operator to the single two-factor model of the portfolio return, and using the zero expectation of residuals, the expectation of the portfolio return is:
E(Y7) = 0.085 + 0.680^) + 1.080£(X,)
The expectation of the portfolio return depends on the constants of each of the factor models. If we use standardized normal factors, return expectations then collapse to zero. It collapses to the weighted constant because each factor has a zero expectation.