The price of risk based on multilinear measures

In this paper, the price of risk measuring how risk and return can be traded off in making portfolio choices is based on multilinear indices that are obtained by using a multilinear and quadratic metric. Since a multilinear and quadratic metric identifying tensors is used, such a price coincides with the Sharpe ratio. It is obtained by studying a multiple random good of order m from a microeconomic point of view. The elements characterizing the Sharpe ratio are accordingly of an objective and subjective nature. They are not of an objective nature only. This is compatible with the fact that risk is intrinsically a notion of a subjective nature connected with the standard deviation of the portfolio return. This paper methodologically studies m risky assets inside of a linear manifold over R. 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 shows that all risky assets contained in an m-dimensional linear manifold are intrinsically related. In particular, any two risky assets of them are 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 take place.