# Existence and Uniqueness for the Multivariate Discrete Terminal Wealth Relative

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Definition of a Terminal Wealth Relative

**Definition**

**1.**

**Lemma**

**1.**

**Proof.**

## 3. Optimal Fraction of the Discrete Terminal Wealth Relative

**Lemma**

**2.**

**Proof.**

**Assumption**

**1.**

- (a)
- $\begin{array}{c}\forall \phantom{\rule{0.166667em}{0ex}}\mathit{\phi}\in \partial {B}_{\epsilon}\left(0\right)\cap {\mathsf{\Lambda}}_{\epsilon}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\exists \phantom{\rule{0.166667em}{0ex}}{i}_{0}={i}_{0}\left(\mathit{\phi}\right)\in \{1,\dots ,N\}\hfill \\ \mathit{such}\phantom{\rule{4pt}{0ex}}\mathit{that}\phantom{\rule{4pt}{0ex}}\langle {({{\mathit{t}}_{{i}_{0}}}_{\xb7}\phantom{\rule{-1.111pt}{0ex}}/\phantom{\rule{-0.55542pt}{0ex}}\widehat{\mathit{t}})}^{\top},\phantom{\rule{4pt}{0ex}}\mathit{\phi}\rangle <0\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\left(\mathit{no}\phantom{\rule{4pt}{0ex}}\mathit{risk}\phantom{\rule{4pt}{0ex}}\mathit{free}\phantom{\rule{4pt}{0ex}}\mathit{investment}\right)\hfill \end{array}$
- (b)
- $\frac{1}{N}\sum _{i=1}^{N}{t}_{i,k}>0\phantom{\rule{1.em}{0ex}}\forall \phantom{\rule{0.166667em}{0ex}}k=1,\dots ,M\phantom{\rule{1.em}{0ex}}\left(\mathit{each}\phantom{\rule{4pt}{0ex}}\mathit{trading}\phantom{\rule{4pt}{0ex}}\mathit{system}\phantom{\rule{4pt}{0ex}}\mathit{is}\phantom{\rule{4pt}{0ex}}\mathit{profitable}\right)$
- (c)
- $ker\left(T\right)=\left\{0\right\}\phantom{\rule{8.em}{0ex}}\left(\mathit{linear}\phantom{\rule{4pt}{0ex}}\mathit{independent}\phantom{\rule{4pt}{0ex}}\mathit{trading}\phantom{\rule{4pt}{0ex}}\mathit{systems}\right)$

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

**Proof.**

**Theorem 2.**

**(optimal f existence)**

- (a)
- ${\mathit{\phi}}_{N}^{opt}$ is unique, or
- (b)
- ${\mathit{\phi}}_{N}^{opt}\in \partial \mathfrak{G}$.

**Proof.**

**Lemma**

**6.**

**Proof.**

**Corollary 1.**

**(optimal f uniqueness)**

**Proof.**

**Remark**

**1.**

## 4. Examples

**Remark**

**2.**

- minimize risk for a given chance/utility level
- or
- maximize chance/utility for a given risk level,

- (a)
- the half spaces for rows 4 and 5 of the return matrix cover the whole set ${\mathbb{R}}_{\ge 0}^{2}$ (cf. Figure 2b),
- (b)
- $\frac{1}{5}{\displaystyle \sum _{i=1}^{5}}{t}_{i,1}=\frac{9}{5}>0$ and $\frac{1}{5}{\displaystyle \sum _{i=1}^{5}}{t}_{i,2}=\frac{6}{5}>0$ and
- (c)
- obviously, the columns of the return matrix are linearly independent.

_{5}with the three trading systems in (19). Corollary 1 yields the uniqueness of this maximal solution for

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Two hyperplanes and the set $\partial {B}_{\epsilon}\left(0\right)\cap {\mathsf{\Lambda}}_{\epsilon}$.

**Figure 3.**Solutions of the linear equations from (17).

**Figure 5.**The Terminal Wealth Relative from Figure 4, view from above.

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Hermes, A.; Maier-Paape, S. Existence and Uniqueness for the Multivariate Discrete Terminal Wealth Relative. *Risks* **2017**, *5*, 44.
https://doi.org/10.3390/risks5030044

**AMA Style**

Hermes A, Maier-Paape S. Existence and Uniqueness for the Multivariate Discrete Terminal Wealth Relative. *Risks*. 2017; 5(3):44.
https://doi.org/10.3390/risks5030044

**Chicago/Turabian Style**

Hermes, Andreas, and Stanislaus Maier-Paape. 2017. "Existence and Uniqueness for the Multivariate Discrete Terminal Wealth Relative" *Risks* 5, no. 3: 44.
https://doi.org/10.3390/risks5030044