Abstract
Monetary risk measures defined on a convex set are interpreted as the smallest amount of external cash that must be added to a portfolio to make the portfolio being acceptable. In the present paper, the authors introduce a new concept: non-cash risk measure, which does as a nonconvex risk measure work in a nonconvex set. In addition, the authors arrive at a convex extension of the non-cash risk measure, and offer the relationship between the non-cash risk measure and its extension.
Highlights
Since Markowitz(1952) [1] put forward the theory of optimal portfolio selection in 1952, variance had become an influential classical financial risk measurement method
It is found that the axioms of positive homogeneity and sub-additivity in coherent risk measure are too strict, and the restrictions of positive homogeneity and sub-additivity should be relaxed
This paper mainly studies how to define the non-cash risk measure and the corresponding risk acceptable set when the costs functions are complicated
Summary
Since Markowitz(1952) [1] put forward the theory of optimal portfolio selection in 1952, variance had become an influential classical financial risk measurement method. This paper mainly studies how to define the non-cash risk measure and the corresponding risk acceptable set when the costs functions are complicated. Non-cash risk measure provides a direct path from unacceptable positions towards the acceptance set. It uses cash and other kinds of assets to adjust the position and return the smallest amount of assets which has to be changed into an eligible quota such that the new position becomes acceptable.
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