Balanced portfolio selection Olivier Cailloux Vincent Mousseau Jun - PowerPoint PPT Presentation

. . . . . . . Balanced portfolio selection Olivier Cailloux Vincent Mousseau Jun Zheng ILLC - Universiteit van Amsterdam October 7, 2013

Introduction . Conclusion 5 . Green labelling 4 . . Mathematical program 3 . Problem Formulation Problem Formulation 2 . . Introduction 1 . . Outline Conclusion Green labelling Mathematical program 2 / 47

Introduction Problem Formulation Conclusion 5 . Green labelling 4 . . Mathematical program 3 . . 2 Problem Formulation . . Introduction 1 . . Outline Literature review Context Conclusion Green labelling Mathematical program 3 / 47

Introduction Student 4 good 5 8 … male Student 3 average 6 4 … female bad female 0 7 … male Student 5 good 4 9 … male … Student 2 … Problem Formulation . Mathematical program Green labelling Conclusion Context Literature review A few examples . Enroll students in a university . . . . 7 Select the “best” 𝑦 students among the candidates Possibly select less than 𝑦 if not enough good students Also: obtain a good balance (e.g. gender balance) “I want those highly motivated and ≥ 8 in literature”? motivation math. literature … gender Student 1 good 9 4 / 47

Introduction . 7k . . . . 5 . . 5 . . 0 Prj 3 . . . AI . . 13k . . . . 2 . . AI 4 . ? . ? . ? . 𝐷 � : reject . 𝐷 � : maybe . 𝐷 � : fund project . . . . . . . . . . . . . … . . . Problem Formulation Fund the “best” proposals exp . sci . redac . . . with a good balance between risk, field… Only fund projects if they are good enough! quality; experience of the team; … Evaluation on multiple criteria: redaction quality; scientific . … . . . . Allocate budget to research proposals . A few examples Literature review Context Conclusion Green labelling Mathematical program . . . . 5 . Prj 2 . . OR . . 10k . . . 5 budget . . 2 . . 3 . Prj 1 . . field . 5 / 47

Introduction Those that improved a lot Category sizes increase (5% best, 10% moderate, …) . . . . . Category sizes . → …however not exactly a ranking → comparisons, as in a ranking them) Also reward those that improved moderately (at most 10% of Only the best of those (at most 5%) Accounting for their improvements on multiple criteria Problem Formulation I want to reward some team members . . . . . Reward improvements (sports team) . A few examples Literature review Context Conclusion Green labelling Mathematical program 6 / 47

Introduction . Classes should be of homogeneous level Classes should be of approximately the same size Split students in language classes . . . . Partition in similar classes Problem Formulation . A few examples Literature review Context Conclusion Green labelling Mathematical program 7 / 47

Introduction . such items Limited patience of the boss/wife/husband limit the number of Intermediary cases: ask the boss (or your wife or husband) KO category: goes to trash OK category: could still be sold (or kept) Sort old items in a shop (or in your attic) . . . Problem Formulation . Yes / No / I don’t know . A few examples Literature review Context Conclusion Green labelling Mathematical program 8 / 47

Introduction . . . . 5 . . 5 . . 0 . Prj 3 7k . . AI . . 13k . . . . 2 . . . 4 . ? . ? . ? . 𝐷 � : reject . 𝐷 � : select . . . . . . . . . . . . . … . . AI . . Problem Formulation No intuitive explanation of the selection . … . exp . sci . redac . . . Only relative value considered . Only two categories Usual approaches use one value per portfolio (no flexibility!) Multicriteria portfolio problem: value is given by multiple criteria satisfying some constraints (knapsack) Classical portfolio selection: select the best subset of items This is a variant of an old problem! Old problem revisited Literature review Context Conclusion Green labelling Mathematical program budget . . . 5 . . Prj 2 . . OR . . 10k . . . field 5 . . 2 . . 3 . . Prj 1 . . 9 / 47

Introduction Preference Theory [Golabi et al., 1981] No explanation about why an alternative is selected Choose best valued portfolio among remaining ones Screen-out portfolio with unsatisfactory balance Screen-out insufficiently good alternatives A portfolio has a value (using value theory) Goal is simply to choose one portfolio Selecting a Portfolio of Solar Energy Projects Using Multiattribute Problem Formulation Excluding with constraints Literature review Context Conclusion Green labelling Mathematical program 10 / 47

Introduction Choose the highest valued portfolio Multiple categories not supported Resulting model does not compare to norms No absolute evaluation Preference model: weights of the attributes Attributes evaluated on the same scale (same distance measure) According to balance on attributes distribution of specific attributes [Farquhar and Rao, 1976] Problem Formulation Evaluating the whole portfolio: a balance model to measure the A balance model Literature review Context Conclusion Green labelling Mathematical program 11 / 47

Introduction Problem Formulation Mathematical program Green labelling Conclusion Context Literature review Robust portfolio selection Combining preference programming with portfolio selection [Liesiö et al., 2007, Liesiö et al., 2008] Robust portfolio selection Screen-out portfolios based on constraints and robust decisions No absolute comparisons Selection not easily explained 12 / 47

Introduction 2 Conclusion 5 . Green labelling 4 . . Mathematical program 3 . . Problem Formulation . Problem Formulation . Introduction 1 . . Outline Method summary Preference elicitation Électre Tri variant General description Conclusion Green labelling Mathematical program 13 / 47

Introduction Alt 3 . . Alt 1 . Alt 2 . . Evaluated using criteria and attributes Alt 4 . 𝐷 � : “Good” . 𝐷 � : “Average” . 𝐷 � : “Bad” (Possibly) A decision to be repeated To be sorted in ordered categories Problem Formulation Method summary Mathematical program Green labelling Conclusion General description Électre Tri variant Preference elicitation Decision situation A set of alternatives . Decision situation . . . . . 14 / 47

Introduction Obtain a sorting function Easily explain assignments the overall portfolio quality (e.g. good balance?) the quality of individuals Select a portfolio using: Judge alternatives on the same ground Comparison of alternatives to norms . . . . . Desirable features of the sorting function . . Problem Formulation . . . . Objective of our method . Objectives Method summary Preference elicitation Électre Tri variant General description Conclusion Green labelling Mathematical program 15 / 47

Introduction 𝐷 � : Bad Alt 2 . … . 𝐷 � : Good . 𝐷 � : Average . . Problem Formulation Preference information considered at two levels . . . . . . . . . Alt 1 . preference model Mathematical program Green labelling Conclusion General description Électre Tri variant Preference elicitation Method summary . Principle of the sorting function . . . . . Uses the alternatives performances on the criteria The sorting function reflects the Decision Maker ( DM ) preferences thanks to a preference model . . 16 / 47 1 Intrinsic alternatives evaluations: “is it good enough? ” 2 Portfolio evaluations: balance? category size? …

Introduction Problem Formulation . . function proceeds . 𝐷 � : Bad . 𝐷 � : Average . 𝐷 � : Good . … . Alt 2 . Alt 1 Mathematical program Green labelling Conclusion General description Électre Tri variant Preference elicitation Method summary Explanation of the method . . preference model . 17 / 47 1 Assuming the preference model is known , explain how the sorting 2 Then: explain how to obtain the preference model

Introduction . Majority threshold: 𝜇 ∈ ℝ ∈ 𝑌 � � . . . . . Preference parameters . Categories: 𝒟 (one category: 𝐷 � , 1 ≤ ℎ ≤ 𝑙 ) Criteria: 𝒦 (one criterion: 𝑘 ) Problem Formulation Alternatives: 𝒝 (one alternative: 𝑏 ) . . Mathematical program Green labelling Conclusion General description Électre Tri variant Preference elicitation Method summary Notations . Objective data . . 18 / 47 Performance of 𝑏 on criterion 𝑘 : 𝑕 � (𝑏) , 𝑕 � ∶ 𝒝 → 𝑌 � ⪰ � , ≻ � defined on 𝑌 � (here: ≥, > ) Lower limit of category 𝐷 � on criterion 𝑘 : 𝑚 � � Weight of criterion 𝑘 : 𝑥 � ∈ ℝ