# Possibilistic investment models with background risk

## TL;DR

This paper explores possibilistic investment models incorporating background risk, using fuzzy numbers and random variables, and derives approximate solutions for portfolio optimization under these hybrid risk representations.

## Contribution

It introduces three novel models combining probabilistic and possibilistic risks in investment problems, with derived approximate solutions for optimal portfolios.

## Key findings

- Developed three hybrid risk models for investment problems.
- Derived approximate formulas for optimal investment solutions.
- Enhanced understanding of combining fuzzy and probabilistic risks.

## Abstract

In the study of investment problem, aside from the investment risk the background risk appears. Both the investment risk and the background risk are probabilistically described by random variables. This paper starts from the hypothesis that the two types of risk can be represented both probabilistically (by random variables) and possibilistically (by fuzzy numbers). We will study three models in which the investment risk and the background risk can be: fuzzy numbers, a random variabl-a fuzzy number and a fuzzy number-a random variable. A portfolio problem is formulated for each model and an approximate calculation formula of the optimal solution is proved.