Proofs for the New Definitions in Financial Markets

TL;DR

This paper introduces new theorems and proofs that extend standard utility theory in finance by redefining risk attitudes independent of utility curve shape, broadening the understanding of investor behavior.

Contribution

It presents novel definitions and proofs that decouple risk attitudes from utility curve shape, expanding the theoretical framework of decision-making under uncertainty in finance.

Findings

Convex utility curves can indicate risk-averse behavior under new definitions.

New definitions are more comprehensive than standard ones when considering utility shape.

Theorems extend standard utility theory by broadening risk attitude classifications.

Abstract

The aim of this study is to present proofs for new theorems. Basic thoughts of new definitions emerge from the decision-making under uncertainty in economics and finance. Shape of the certain utility curve is central to standard definitions in determining risk attitudes of investors. Shape alone determines risk behavior of investors in standard theory. Although the terms risk-averse, risk-loving, and risk-neutral are equivalent to strict concavity, strict convexity, and linearity, respectively, in standard theory, strict concavity or strict convexity, or linearity are valid for certain new definitions. The connection between the curvature of utility curve and risk attitude is broken for the new definitions. For instance, convex utility curve may show risk-averse behavior under new definitions. Additionally, this paper has proved that new definitions are richer than standard ones when shape is considered. Hence, it can be stated that new definitions are broader than standard definitions from the viewpoint of shape. With all of these, it has been demonstrated that the theorems and proofs in this study extend the standard utility theory in an important way.