# The Effect of Prudence on the Optimal Allocation in Possibilistic and Mixed Models

## Abstract

**:**

## 1. Introduction

- a bidimensional probabilistic expected utility, when both components are random variables;
- a bidimensional possibilistic expected utility ([16], p. 60), when both components are fuzzy numbers;
- a mixed expected utility ([16], p. 79), when a component is a random variable, and the other is a fuzzy number.

## 2. Preliminaries

#### 2.1. Possibilistic Expected Utility

- a utility function u of class ${\mathcal{C}}^{2}$,
- a fuzzy number A whose level sets are ${[A]}^{\gamma}=[{a}_{1}(\gamma ),{a}_{2}(\gamma )]$, $\gamma \in [0,1]$,
- a weighting function $f:[0,1]\to \mathbf{R}$. (f is a non-negative and increasing function that satisfies ${\int}_{0}^{1}f(\gamma )d\gamma =1$).

**Proposition**

**1.**

**Corollary**

**1.**

#### 2.2. Mixed Expected Utility

- In the first step, the possibilistic risk is parametrized by the decomposition of A in its level sets $[{a}_{1}(\gamma ),{a}_{2}(\gamma )],\gamma \in [0,1]$.
- In the second step, for each level $\gamma $ one considers the parametrized probabilistic utilities $M(u({a}_{1}(\gamma ),X))$ and $M(u({a}_{2}(\gamma ),X))$.
- In the third step, the mixed expected utility ${E}_{f}(u(A,X))$ is obtained as the f-weighted average of the family of means$$(\frac{1}{2}{[M(u({a}_{1}(\gamma ),X))+M(u({a}_{2}(\gamma ))])}_{\gamma \in [0,1]}.$$

**Remark**

**1.**

**Proposition**

**2.**

**Corollary**

**2.**

## 3. Possibilistic Standard Model

- In (8) there is a probabilistic risk X, and in (10) there is a possibilistic risk A.
- Problem (8) is formulated in terms of a probabilistic expected utility operator $M(u(.))$, while (10) is formulated using the possibilistic expected utility operator ${E}_{f}(u(.))$.

## 4. The Effect of Prudence on the Optimal Allocation

**Proposition**

**3.**

**Proposition**

**4.**

**Theorem**

**1.**

**Remark**

**2.**

**Example**

**1.**

## 5. Models with Background Risk

## 6. Approximate Solutions of Portfolio Choice Model with Background Risk

**Proposition**

**5.**

**Proposition**

**6.**

**Theorem**

**2.**

**Remark**

**3.**

**Example**

**2.**

**Theorem**

**3.**

**Example**

**3.**

**Theorem**

**4.**

**Example**

**4.**

- B is a triangular fuzzy number $B=(b,\alpha ,\beta )$ and C is a symmetric triangular fuzzy number $C=(c,\delta )$,
- the utility function u is of HARA-type: $u(w)=\zeta {(\eta +\frac{w}{\gamma})}^{-1}$ for $\eta +\frac{w}{\gamma}>0$,
- the weighting function f has the form $f(t)=2t$ for $t\in [0,1]$.

**Theorem**

**5.**

**Example**

**5.**

- X has the normal distribution $N(m,\sigma )$ and C is the triangular fuzzy numbers $C=(c,\delta ,\u03f5),$
- the utility function u is of HARA-type: $u(w)=\zeta {(\eta +\frac{w}{\gamma})}^{-1}$ for $\eta +\frac{w}{\gamma}>0$,
- the weighting function f has the form: $f(t)=2$ for $t\in [0,1]$.Then, $M(X)=m$, $Var(X)={\sigma}^{2}$, $M[{(X-M(X))}^{3}]=0$, and ${E}_{f}(c)=c+\frac{\u03f5-\delta}{6}$.It follows the following form of ${\beta}_{3}(k)$:$${\beta}_{3}(k)\approx \frac{m}{{\sigma}^{2}}[\frac{1}{{r}_{u}(w)}-{E}_{f}(C)]=\frac{m}{{\sigma}^{2}}[\eta +\frac{w}{\gamma}-c-\frac{\u03f5-\delta}{6}].$$

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Proof**

**of**

**Corollary**

**2.**

**Proof**

**of**

**Proposition**

**3.**

**Proof**

**of**

**Proposition**

**4.**

**Proof**

**of**

**Theorem**

**1.**

**Proof**

**of**

**Proposition**

**5.**

**Proof**

**of**

**Proposition**

**6.**

**Proof**

**of**

**Theorem**

**2.**

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Investment Risk | Background Risk | |
---|---|---|

1 | probabilistic | probabilistic |

2 | possibilistic | possibilistic |

3 | possibilistic | probabilistic |

4 | probabilistic | possibilistic |

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**MDPI and ACS Style**

Georgescu, I.
The Effect of Prudence on the Optimal Allocation in Possibilistic and Mixed Models. *Mathematics* **2018**, *6*, 133.
https://doi.org/10.3390/math6080133

**AMA Style**

Georgescu I.
The Effect of Prudence on the Optimal Allocation in Possibilistic and Mixed Models. *Mathematics*. 2018; 6(8):133.
https://doi.org/10.3390/math6080133

**Chicago/Turabian Style**

Georgescu, Irina.
2018. "The Effect of Prudence on the Optimal Allocation in Possibilistic and Mixed Models" *Mathematics* 6, no. 8: 133.
https://doi.org/10.3390/math6080133