Theoretically Scrutinizing Kinks on Efficient Frontiers and Computationally Reporting Nonexistence of the Tangent Portfolio for the Capital Asset Pricing Model by Parametric-Quadratic Programming

Researchers traditionally compute isolated points for an efficient frontier and assume a line which passes through a risk-free asset rf and is tangent to the frontier. The tangency plays pivotal roles for the capital asset pricing model (CAPM). However, the assumption may not hold in the presence of kinks (as non-differentiable points) on efficient frontiers. Kinks are detected by parametric-quadratic programming only and not by ordinary portfolio optimization. Up until now, there has been no research to theoretically scrutinize kink properties (especially implications to CAPM) and systematically quantify the nonexistence of the tangency. In such an area, this paper contributes to the literature. In theorems and corollaries, we prove the nonexistence of the tangency and substantiate that expected-return axis is composed of piecewisely connected intervals for which the tangency does not exist and intervals for which the tangency exists. Computationally, we reveal universal existence of kinks (e.g., 0.2 to 8.0 kinks for 5-stock to 1800-stock portfolio selections) and the tangency-nonexistence ratios as about 0.066.