Existence, uniqueness and stability of optimal portfolios of eligible assets

TL;DR

This paper investigates the existence, uniqueness, and stability of optimal portfolios of eligible assets in capital adequacy frameworks, emphasizing the importance of continuity properties for financial applications.

Contribution

It provides new insights into the conditions ensuring the stability and lower semicontinuity of optimal portfolios, especially beyond polyhedral risk measures.

Findings

Optimal portfolios exist and are unique under certain conditions.

Lower semicontinuity holds for polyhedral risk measures but may fail otherwise.

Focusing on near-optimal portfolios can restore stability.

Abstract

In a capital adequacy framework, risk measures are used to determine the minimal amount of capital that a financial institution has to raise and invest in a portfolio of pre-specified eligible assets in order to pass a given capital adequacy test. From a capital efficiency perspective, it is important to identify the set of portfolios of eligible assets that allow to pass the test by raising the least amount of capital. We study the existence and uniqueness of such optimal portfolios as well as their sensitivity to changes in the underlying capital position. This naturally leads to investigating the continuity properties of the set-valued map associating to each capital position the corresponding set of optimal portfolios. We pay special attention to lower semicontinuity, which is the key continuity property from a financial perspective. This "stability" property is always satisfied if the test is based on a polyhedral risk measure but it generally fails once we depart from polyhedrality even when the reference risk measure is convex. However, lower semicontinuity can be often achieved if one if one is willing to focuses on portfolios that are close to being optimal. Besides capital adequacy, our results have a variety of natural applications to pricing, hedging, and capital allocation problems.