# A Multicriteria Extension of the Efficient Market Hypothesis

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## Abstract

**:**

## 1. Introduction

- A theorem connecting normalization techniques and p-norm transformations;
- A theorem providing a sufficient condition for the percentile normalization to be equivalent to the linear max–min normalization.

## 2. Fama’s Formulation of the Efficient Market Hypothesis and an Empirical Test

## 3. A Multidimensional Approach to the EMH

#### 3.1. Critical Dimensions to Test the EMH

- A n-dimensional set of criteria functions $\mathcal{G}=\{{g}_{1}(\xb7),\dots ,{g}_{j}(\xb7),\dots ,{g}_{n}(\xb7)\}$;
- An algorithm $\alpha \left({\Phi}_{t}\right)$ that determines a candidate portfolio ${\mathit{x}}_{t}$ at each time step t based on a given information sequence ${\Phi}_{t}$ indexed by $t\in \{1,2,\dots ,m\}$;
- A benchmark portfolio ${\mathit{b}}_{t}$ that can also be evaluated in terms of set $\mathcal{G}$ defining the equilibrium performance of the market;
- A dissimilarity measure $\mu (A,B)$, where A and B are $m\times n$ data matrices with elements set to ${a}_{tj}={g}_{j}\left({\mathit{x}}_{t}\right)$ and ${b}_{tj}={g}_{j}\left({\mathit{b}}_{t}\right)$, respectively.

- Number of criteria under consideration.
- The time horizon (sample size).
- A dissimilarity measure.
- A summary result to prove or disprove the EMH.

#### 3.2. Basic Requirements

**Exhaustiveness**. The set of criteria must minimize the loss of information that any model implies with respect to the reality that it is trying to represent;**Cohesiveness**. The set of criteria must ensure compatibility between the role that each criterion plays and the more comprehensive role that a set of criteria plays when integrating all preferences. Cohesiveness implies that if decision a results in degrading one criterion and decision b results in improving another criterion, then b must outrank a with respect to comprehensive preferences;**Non-redundancy**. A set of criteria is non-redundant when leaving out some criterion implies the infringement of either exhaustiveness or cohesiveness.

**Consistency**. We argue that the reliability of an EMH test should be proportional to the number of criteria under consideration, but also to the size of the dataset used. Thus, we here introduce the concept of (time, criteria)-consistency of the test, or $(m,n)$-consistency, according to some dissimilarity measure $\mu (A,B)$, to describe the power of the test. An EMH test based on returns and risk is more consistent than another EMH based only on returns. This reasoning is behind the usual risk adjustment in regular EMH tests. Similarly, EMH tests considering additional criteria such as diversification, liquidity, dividends, social and environmental responsibility, and the amount of short selling, are more criteria-consistent, provided that exhaustiveness, cohesiveness and non-redundancy are respected. Furthermore, an EMH test based on larger datasets (covering longer periods of time and, ideally, including bull-market periods, bear-market conditions and sideways trends) are more time-consistent than other tests based on smaller datasets;**Compensability**. The selection of the aggregation of multiple criteria may have an impact in terms of the compensation among criteria. Here, compensability means the degree to which the achievement of any criterion is offset or balanced by the achievement of another one. This concept is a critical issue when categorizing aggregation methods such as the one described in this paper [15,16,17];**Robustness**. The presence of outliers or measurement errors in the data used to test the EMH may lead to distorted or wrong results. Even though (time-criteria)-consistency may significantly contribute to reducing the impact of otuliers in a multidimensional EMH test due to the need for larger datasets, methods and dissimilarity measures that are more robust to the presence of outliers are preferred.

## 4. Main Steps of a Multidimensional EMH Test

- Normalize criteria;
- Design a weight system;
- Define a dissimilarity measure;
- Determine the result for the test.

#### 4.1. Normalize Criteria

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

#### 4.2. Design a Weight System

#### 4.3. Define a Signed Dissimilarity Measure

**Definition**

**1.**

**Definition**

**2.**

- 1.
- (Non-negativity) For all element $E\in \Sigma $, we have that $\mu \left(E\right)\ge 0$;
- 2.
- (Null-empty set) Measure μ over the empty set is null, $\mu (\varnothing )=0$;
- 3.
- (Countable additivity) For all countable collections ${\left\{{E}_{k}\right\}}_{k=1}^{\infty}\in \Sigma $ of pairwise disjoint sets, it holds that$$\mu \left(\bigcup _{k=1}^{\infty}{E}_{k}\right)=\sum _{k=1}^{\infty}\mu \left({E}_{k}\right).$$

**Definition**

**3.**

**Definition**

**4.**

#### 4.4. Determine the Result of the EMH Test

## 5. Illustrative Example

- The SPDR EURO STOXX 50 ETF seeks to provide investment results that correspond to the performance of the EURO STOXX 50 Index as a benchmark. The EURO STOXX 50 Index is designed to represent the performance of some of the largest companies across components of the 19 EURO STOXX supersectors in terms of capitalization of 11 European countries;
- The iShares STOXX Europe 600 ETF (DE) seeks to track the performance of the EURO STOXX 600 Index as a benchmark. The EURO STOXX 600 Index is designed to represent the performance of a fixed number of 600 components representing large, mid and small capitalization companies among 17 European countries.

- Weekly return (WR), computed as the difference in value during 6 days$$WR=\frac{{V}_{t}-{V}_{t-6}}{{V}_{t-6}}.$$
- Compound annual growth rate (CAGR), as a proxy a constant rate of return over a given time period$$CAGR={\left(\right)}^{\frac{{V}_{t}}{{V}_{t-6}}}6$$

- Sample standard deviation (SD) of six daily returns grouped by weeks$$SD={\left(\right)}^{\frac{1}{5}}1/2$$
- Maximum drawdown (MD) is the peak-to-trough decline in value during a week$$MD=\frac{{V}_{tL}}{{V}_{tP}}-1$$

#### 5.1. Alignment and Normalization

#### 5.2. Neutral Weight System

#### 5.3. Defining Eight Different Dissimilarity Measures

- Case 1. Considering only a measure of return ($WR$);
- Case 2. Considering a measure of return ($WR$) and a measure of risk ($SD$);
- Case 3. Considering two measures of return ($WR$ and $CAGR$) and a measure of risk ($SD$);
- Case 4. Considering two measures of return ($WR$ and $CAGR$) and two measures of risk ($SD$ and $MD$).

#### 5.4. Results of the EMH Test

## 6. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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${\mathit{d}}_{\mathit{tj}}$ | p | $\mathbf{sign}({\mathit{d}}_{\mathit{tj}}^{\mathit{p}})$ | ${\mathbf{sign}}^{\mathit{p}+1}\left({\mathit{d}}_{\mathit{tj}}\right)$ | ${\mathbf{sign}}^{\mathit{p}+1}\left({\mathit{d}}_{\mathit{tj}}\right)\xb7{\mathit{d}}_{\mathit{tj}}^{\mathit{p}}$ |
---|---|---|---|---|

${d}_{tj}\ge 0$ | 0, 2, 4, … | 1 | 1 | 1 |

${d}_{tj}<0$ | 0, 2, 4, … | 1 | −1 | −1 |

${d}_{tj}\ge 0$ | 1, 3, 5, … | 1 | 1 | 1 |

${d}_{tj}<0$ | 1, 3, 5, … | −1 | 1 | −1 |

Case 1 | Case 2 | Case 3 | Case 4 | |
---|---|---|---|---|

$\Delta (\mathit{A},\mathit{B},\mathit{p})$ | $\left(\mathit{WR}\right)$ | $(\mathit{WR};\mathit{SD})$ | $(\mathit{WR};\mathit{SD};\mathit{CAGR})$ | $(\mathit{WR};\mathit{SD};\mathit{CAGR};\mathit{MD})$ |

$p=1$ | 8.6% | −109.4% | −49.6% | −75.5% |

$p=0$ | 24.8% | −80.5% | −42.0% | −68.6% |

Case 1 | Case 2 | Case 3 | Case 4 | |
---|---|---|---|---|

$\Delta (\mathit{A},\mathit{B},\mathit{p})$ | $\left(\mathit{WR}\right)$ | $(\mathit{WR};\mathit{SD})$ | $(\mathit{WR};\mathit{SD};\mathit{CAGR})$ | $(\mathit{WR};\mathit{SD};\mathit{CAGR};\mathit{MD})$ |

$p=1$ | 39.7% | 4.3% | 7.3% | 204.6% |

$p=0$ | 40.2% | 26.7% | 28.4% | 41.6% |

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**MDPI and ACS Style**

Salas-Molina, F.; Pla-Santamaria, D.; Mayor-Vitoria, F.; Vercher-Ferrandiz, M.L.
A Multicriteria Extension of the Efficient Market Hypothesis. *Mathematics* **2021**, *9*, 649.
https://doi.org/10.3390/math9060649

**AMA Style**

Salas-Molina F, Pla-Santamaria D, Mayor-Vitoria F, Vercher-Ferrandiz ML.
A Multicriteria Extension of the Efficient Market Hypothesis. *Mathematics*. 2021; 9(6):649.
https://doi.org/10.3390/math9060649

**Chicago/Turabian Style**

Salas-Molina, Francisco, David Pla-Santamaria, Fernando Mayor-Vitoria, and Maria Luisa Vercher-Ferrandiz.
2021. "A Multicriteria Extension of the Efficient Market Hypothesis" *Mathematics* 9, no. 6: 649.
https://doi.org/10.3390/math9060649