Testing Whether Transformed Asset Returns Fit a One-Factor Model

Summary

The document sets up two asset returns driven by a common factor with opposite loadings, plus uncorrelated residuals of equal variance. It then defines one portfolio as the first asset and another as a weighted combination of both, with the weight tied to the residual variance. The question asks whether this transformation prevents the resulting returns from satisfying a one-factor model with the original factor, and whether they could instead be modeled with zero factors and uncorrelated residuals.

This is a model-testing question rather than a worked analysis: it provides assumptions and definitions but no proof, computed covariances, or answer. The setup highlights that transformed portfolios can have different factor exposures and residual dependence from their constituent assets. Any conclusion requires checking the transformed returns’ covariance structure against the stated model restrictions; the document itself does not establish whether either model holds.

Key ideas

• The example gives two returns opposite loadings on a shared factor and equal residual variances.
• A weighted portfolio transformation may alter both factor exposures and residual covariances.
• A one-factor model can be assessed by checking whether the transformed covariance structure meets its restrictions.
• The document poses the test but provides no derivation or conclusion.

Tags

# How can I show that these assets do not satisfy a 1-factor model?


# How can I show that these assets do not satisfy a 1-factor model?












Suppose these two assets satisfy a 1-factor model: $$ R_1= E(R_1) + F + ε_1 \\ R_2= E(R_2) - F + ε_2 \\ $$ where: $$ E(F)=E(ε_1)=E(ε_2)=0 \\ Var(F)=1, Cov(F,ε_1)=Cov(F,ε_2)=Cov(ε_1,ε_2)=0 \\ Var(ε_1)=Var(ε_2)=\sigma^2 \\ $$ If: $$ S_1=R_1 \\ S_2= \psi R_1 + (1-\psi)R_2 \\ \psi= 1/(2+\sigma^2) $$ How can I show that $S_1$ and $S_2$ do not satisfy a 1-factor model with factor F? [DONE]

Also i wanted to ask if $S_1$ and $S_2$ satisfy the following zero-factor model: $$ S_1= E(S_1) + \widetilde{\epsilon_1} \\ S_2= E(S_2) + \widetilde{\epsilon_2} $$ where the variances of the epsilons and their covariance is zero.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.