Model Uncertainty, Complexity and Rank in Finance

Abstract

There are three crucial mathematical system concepts in Finance, which are either being confused or misapplied - uncertainty, complexity and rank. First, the concept of epistemic uncertainty is sufficient for modeling and the concept of probability is unnecessary. This is illustrated by Galton's Error, and the under-repesentation of systematic risk by American mutual funds. These funds use simple unidirectional projection (regression) to compute Sharpe's beta for fund selection. There are at least five equivalent ways of representing the measured model uncertainty and a new and an improved risk categorization for mutual funds is presented. Second, the concept of (linear) system complexity is usually dealt with by presuming a model rank, as the Cowles Foundation erroneously prescribed in the early 1950s, and superimposing that model rank on the data, when a model is estimated. But the model rank does not have to be presumed: it can be identified from the data and all corresponding (Grassmanian) coefficients can be computed by CLS Projections. This is illustrated by the identification of the model rank of simple financial risk systems in six Asian countries, in particular in Taiwan. Third, often it is thought that Markowitz' portfolio optimization and exact and complete cash flow accounting are incompatible because of the non-existence, or empirical instability of the information matrix. The problem is caused by the rank constraints imposed by the portfolio accounting identities. But these rank constraints also provide the solution, since they form exact selectors of the portfolio allocations, which are found by simple tensor algebra. This will be illustrated by the optimization of an Asian multi-currency stock investment portfolio.