Abstract
This chapter presents mathematical tools that make the algebra of multivariate higher moments very tractable. Based on them, a multidimensional portfolio frontier is created, incorporating skewness and using kurtosis instead of variance. Most models in finance are based on mean-variance analysis. Therefore, risk premium is derived from the second moment of a random variable. The basic assumption of this kind of modeling is that agents are not so concerned about moments higher than the variance. In fact, all the utility functions that are strictly increasing and concave have expected utilities that increase with odd moments and decrease with even moments. When restricting ourselves to expected utility, mean-variance analysis may be justified by either a quadratic utility function or by a Taylor approximation of a more general and acceptable utility function. Differently from the mean-variance case, which only requires linear algebra, higher moments require the use of the so-called tensors, which can be seen as a generalization of matrices.
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