Multi-Time and Multi-Moment Nonparametric Frontier-Based Fund Rating: Proposal and Buy-and-Hold Backtesting Strategy 

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Single-Time and Multi-Moment Rating Framework

The nonparametric frontier rating methods gauge the financial performance of MF, and these evaluations are done mostly using frontier-based models which originate from production theory. In this section, we only introduce the basic definitions and properties needed for applications within finance. Assume that there are n MFs under evaluation over a given time horizon. At time t in this time horizon, the j-th MF (j ∈ {1, . . . , n}) is characterized by m input-like values xtij (i ∈ {1, . . . , m}) and s output-like values yrjt (r ∈ {1, . . . , s }). Input-like variables need to be minimized and output-like variables need to be maximized. We introduce one widely used production frontier-based model with variable returns to scale (VRS). Following Briec, Kerstens, and Vanden Eeckaut (2004), a unified algebraic representation of convex and nonconvex production possibility sets (PPS) under the VRS assumption for a sample of n MFs at time t is: n PΛt = (xt, yt) ∈ Rm × Rs | ∀i ∈ {1, . . . , m} : xit ≥ λjxijt, j=1 n ∀r ∈ {1, . . . , s} : yrt ≤ λjyrjt, λ ∈ Λ , (2.2.1) j=1 where: ≥ 0} if convexity is ∈ {0, 1}} if noncon- Λ ≡ ΛC = {λ ∈ Rn | assumed, and Λ ≡ ΛNC = {λ ∈ Rn | vexity is assumed. n λj = 1 and ∀j ∈ {1, . . . , n} : λj j=1 n λj = 1 and ∀j ∈ {1, . . . , n} : λj j=1 At time t, if there exists an input-output combination ( nj=1 λjxtij, nj=1 λjyijt) in the convex or nonconvex PPS using less inputs and producing more outputs than the observed MF, then this MF is considered inefficient since it can improve its inputs and/or outputs. MFs are efficient if no improved input-output combinations can be found. The input-output combinations of these efficient MFs are all located at the boundary of PΛt which is called the convex or nonconvex VRS nonparametric frontier. Using the nonparametric PPS defined in (2.2.1), the shortage function of any observed MF at time t is now defined as follows: Definition 2.2.1. At time t, let gt = (−gxt, gyt) ∈ Rm− × Rs+ and gt ≠ 0. For any observation zt = (xt, yt) ∈ Rm × Rs, the shortage function SΛt at time t in the direction of vector gt is defined as.

Multi-Time and Multi-Moment Rating Framework

Differing from MF ratings in a single-time framework, MF ratings in a multi-time framework consider performance over a time horizon consisting of multiple discrete time periods. To develop the nonparametric frontier rating models in this multi-time framework, some definitions and properties are presented. Consider n MFs under evaluation. Let T denote the number of consecutive times in a time horizon of interest. In addition, define a multi-time path of inputs and outputs as Zj = (xtj, yjt)Tt=1 for MF j, (j = 1, . . . , n), where xtj = (xt1j, . . . , xtmj) and yjt = (y1tj, . . . , ysjt) represent m inputs and s outputs at time t, respectively.
Assuming VRS for all times t ∈ {1, . . . , T } and strong free disposability of all inputs and outputs, the multi-time PPS with convexity and nonconvexity can be defined as: PΛT = PΛ1 × · · · × PΛT ⊂ (Rm × Rs)T ∼= Rm×T × Rs×T , (2.2.3) where PΛt, (t = 1, . . . , T ), is the PPS at time t mentioned previously in (2.2.1). The idea is now for each MF to simultaneously expand its multiple outputs and decrease its multiple inputs over all discrete times in a given time horizon by means of the multi-time shortage function. To allow a general definition, we first introduce some abbreviating notations.
The time dependent direction vector denoted by G = (g1, . . . , gT ) ∈ (Rm− × Rs+)T ∼= Rm−×T × Rs+×T represents a given multi-time direction path, where gt = (−gxt, gyt) ∈ Rm− × Rs+ represents the direction vector at time t ∈ {1, . . . , T }. In addition, we denote Θ = (β1, . . . , βT ) ∈ RT and Θ · G = (β1g1, . . . , βT gT ) ∈ (Rm × Rs)T ∼= Rm×T × Rs×T . Considering the time preference of an investor in a portfolio context, we introduce a time discounting factor denoted ξ (0 < ξ < 1) to weight the efficiency measures over the time horizon.10 Then, the time discounted multi-time shortage function assuming convexity or nonconvexity is defined as follows:

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Backtesting Framework

Our main objective in this contribution is to test that the multi-time and multi-moment performance measures can be expected to perform well for MF ratings and selection. To this end, a comparative approach based on a backtesting meth-odology is adopted. Backtesting refers to executing fictitious investment strategies using historical data to simulate how these strategies would have performed if they had actually been adopted by MF managers in the past. It is powerful for evaluat-ing and comparing the performance of different investment strategies without using real capital. Some examples of a backtesting approach are found in DeMiguel, Gar-lappi, and Uppal (2009), Tu and Zhou (2011), Brandouy, Kerstens, and Van de Woestyne (2015), Zhou, Xiao, Jin, and Liu (2018) and Lin and Li (2020), among.
For comparison, there are 15 fund rating methods in total being collected in our work. On the one hand, we test some traditional financial indicators: Sharpe ratio, Sortino ratio and Omega ratio. The exact definition for the Sharpe, Sortino and Omega ratios can be found in Feibel (2003, p. 187 and p. 200) and Eling and Schuhmacher (2007, p. 2635), respectively. Based on these definitions, these three traditional financial ratios are presented as follows:

Table of contents :

Acknowledgements
General Abstract
Résumé Général
List of Figures
List of Tables
1 General Introduction 
2 Multi-Time and Multi-Moment Nonparametric Frontier-Based Fund Rating: Proposal and Buy-and-Hold Backtesting Strategy
2.1 Introduction
2.2 Nonparametric Frontier Rating Models: Methodology
2.2.1 Single-Time and Multi-Moment Rating Framework
2.2.2 Multi-Time and Multi-Moment Rating Framework
2.3 Backtesting Framework
2.4 Empirical Backtesting Results
2.4.1 Sample Description
2.4.2 Evaluation Results
2.4.3 Backtesting Results
2.5 Conclusion
2.A Sample Description: Further Details
2.B Backtesting Results: Sensitivity Analysis
3 Multi-Time and Multi-Moment Nonparametric Frontier-Based Fund
Rating: Luenberger Portfolio Productivity Indicator Approach 
3.1 Introduction
3.2 Nonparametric Frontier Rating in Dynamic Analysis Framework: Methodology
3.2.1 Single-Time and Multi-Moment Luenberger Portfolio Productivity Indicator
3.2.2 Multi-Time and Multi-Moment Luenberger Portfolio Productivity Indicator
3.3 Backtesting Framework in Dynamic Analysis
3.4 Empirical Testing
3.4.1 Evaluation Results Comparison
3.4.2 Backtesting results
3.5 Conclusion
4 Multi-Time and Multi-Moment Nonparametric Frontier-Based Fund
Rating: A Nonconvex Metafrontier Approach 
4.1 Introduction
4.2 Portfolio Metafrontier and Nonparametric Estimators: Graphical Illustration
4.3 Nonparametric Metafrontier Rating: Methodology
4.3.1 Group-specific Technology and Group-specific Shortage Function
4.3.2 Metatechnology and Metatechnology Shortage Function
4.3.3 Technology Efficiency and Technology Difference Gap
4.4 Empirical Illustration
4.4.1 Sample Description
4.4.2 Evaluation Results
4.4.3 Backtesting Analysis
4.5 Conclusion
4.A Additional Table and Figures
5 General Conclusion

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