The price of risk based on multilinear measures

This paper studies m risky assets inside of a metric space over R. They generate a distribution of mass of a multiple risky asset of order m. An m-dimensional linear manifold is generated by m basic risky assets. They identify m finite partitions, where each of them is characterized by n mutually exclusive elementary events, with n > m. Given m risky assets, this research work innovatively proves that all risky assets contained in an m-dimensional linear manifold are related. In particular, any two risky assets of them can be α-orthogonal, so their covariance is equal to 0. This paper reinterprets principal component analysis by showing that the principal components are basic risky assets of an m-dimensional linear manifold. Non-classical inferential results are obtained. They prove that constants of riskiness exist. The price of risk measuring how risk and return can be traded off in making portfolio choices is based on multilinear indices. All of this holds with regard to choices being made by the decision-maker under conditions of certainty