The higher-rank numerical range is a convex compact set generalizing the classical numerical range of a square complex matrix, first appearing in the study of quantum error correction. We will discuss some of the real algebraic and convex geometry of these sets, including a generalization of Kippenhahn's theorem, and describe an algorithm to explicitly calculate the higher-rank numerical range of a given matrix.

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Abstract:We examine two interrelated issues in risk-adjusted return on capital performance measurement: estimating hurdle rates and allocating capital to debt instruments in a portfolio. We consider a methodology to differentiate hurdle rates for individual debt instruments that incorporates obligor-specific information. These instrument-specific hurdle rates, which define the required compensation of the shareholders, enable a granular differentiation of systematic risk among debt contracts. Using the proposed approach, we show that the hurdle rate could be materially different among industry sectors and obligors of different credit quality. Profitability assessment could be significantly distorted if the difference in hurdle rates is ignored.JEL classification: G12; G13; G21; G22; G32