Finance and insurance

FEDERICA RICCA

The course introduces students to the main methodologies of constrained, continuous, and discrete optimization. It covers theoretical models for formulating decision problems as optimization problems, with particular emphasis on algebraic models of mathematical programming. The course also presents applications of optimization tools for solving decision-making problems, especially in portfolio selection, including practical laboratory sessions where students solve real economic and financial problems using specialized optimization software. By the end of the course, students will understand the major classes of continuous and discrete optimization models, particularly those most relevant for economic and financial applications. They will be able to identify decision problems suitable for an optimization approach, formulate appropriate model assumptions, and apply optimization software to implement models and obtain solutions. Students will interpret results correctly, critically assess their validity, and discuss their significance in real-world contexts. They will be able to present models and results rigorously using formal technical language. The combination of theoretical learning and practical application will develop students’ autonomy in problem analysis, systematic study methods, and the ability to independently learn new or advanced topics. Upon completion, students will possess a solid quantitative knowledge base and a formal approach to problem analysis, enabling autonomous learning for further studies or professional applications.

The course presents the main optimization models and methodologies for constrained, unconstrained, continuous, discrete and network-based problems, as well as, the necessary tools to formalize a decision problem through an optimization model. In the theoretical lessons, students acquire the formal foundations of mathematical modeling and optimization for the quantitative study of financial problems, with a particular focus on portfolio optimization. In the practical lessons, students learn how to apply optimization models to real-world economic and financial problems using specific optimization software.

1) Knowledge and understanding. By the end of the course, students will know the theory of optimization and mathematical programming and the main optimization models used for financial decision-making, particularly for portfolio selection in the mean-variance framework.

2) Applying knowledge and understanding. Students will be able to address a wide range of practical financial problems using a rigorous modeling approach and computational solutions, employing calculation software. They will be able to select the most appropriate model for each specific problem and evaluate solutions, including from a multi-criteria perspective.

3) Making judgments. Based on their knowledge of the variety of models presented and their different abilities to capture the essential aspects of a problem, students will develop critical thinking and rigorous reasoning about the relationships between available models and real-world problems. They will be able to structure a problem and identify the essential elements to be included in its mathematical representation.

4) Communication skills. To address problems using quantitative models, students must acquire an appropriate formal language. They will discuss models in group work and present the concepts and topics covered in the course. This practice, guided by the instructor, encourages students to write algebraic models and present them orally, thereby developing mastery of formal technical language and the ability to explain models to non-technical audiences clearly, precisely, and rigorously.

5) Learning skills. During the course, students will be encouraged to conduct independent research, supported by the instructor when necessary, consulting reference literature and reviewing specialized publications in the field. Students will be able to continue and independently develop their studies in this context and, more generally, will acquire the ability to perform bibliographic research and in-depth investigations on quantitative topics of interest, which will also be valuable for preparing their thesis.

Basic algebra – set theory essentials – equalities and inequalities – systems of linear equations – basics of Statistics.

1. Optimization models and techniques: Optimization and Mathematical Programming – Linear Programming (LP) – Geometrical interpretation of a Linear Program – The Simplex Method – Essentials of Duality in Linear Programming.
Integer and Mixed Integer Linear Programs (ILP, MILP) – Essentials in solution algorithms for ILP and MILP – Essentials of multi-objective optimization. Essentials of Graphs and Networks and Network flow optimization models.

2. Applications in finance: Asset Liability models – Capital Budgeting – Portfolio Selection models: risk-return models, models based on stochastic dominance, risk diversification models, index tracking.

3. Computational Finance with Excel and Matlab: Introduction to Matlab – Financial data preparation and processing – Optimization tools in Excel and Matlab – Practical solution of financial models.

G. Cornuejols, J. Pena, R. Tutuncu (2018) – Optimization Methods in Finance, Cambridge Univ. Press, 2nd edition;
P. Rardin, Optimization in Operations Research, Upper Saddle River, Prentice-Hall, 1998.
J. Elton and J. Gruber (2013), Modern Portfolio Theory and investment analysis, John Wiley and Sons (ISBN-13: 978-1118469941)

Teacher's handouts.

COURSE MATERIAL
Slides and materials are available on the e-learnig platform Moodle:
Link: https://elearning.uniroma1.it/course/view.php?id=13515
Course registration key: FinOpt_21-22

F. Cesarone (2020), Computational Finance. MATLAB oriented modeling, Routledge-Giappichelli Studies in Business and Management (ISBN-13: 978-8892132504)
G.L. Thompson, S. Thore (1992), Computational Economics, The Scientific Press (ISBN-13: 978-1578261857)

The course (worth 6 ECTS credits) is structured by alternating classroom lectures with practical sessions in a computer lab. For each theoretical/methodological topic presented during lectures, a corresponding lab session is dedicated to practical applications for solving financial optimization problems using a Solver (Excel/Matlab). In these sessions, students work on model formulation exercises, implement and solve them, and critically discuss the solutions in terms of their validity within the application context or the need to revise the model that generated them.
In the first part of the course, lectures focus on illustrating the main optimization methods and models, along with the tools needed to learn how to formulate and analyze them. These are presented through small-scale educational examples, always within the context of economic and financial problems. Around the midpoint of the course, once students have acquired the basic tools, lectures shift to presenting more advanced optimization models that are relevant for studying a wide range of real-world financial problems. These applied cases form the basis for lab activities, where students are assigned group tasks developed collaboratively among peers, under the supervision of the instructor and with the possibility of consulting them when needed.
In addition to in-class practical activities, students are also assigned independent tasks to be completed outside of class. For each assignment, students are given sufficient time to complete the work, and the solutions are then presented and discussed in a subsequent lesson. The practical activities aim to develop students’ independent analytical and problem-solving skills through the application of optimization models. To consolidate and assess the skills acquired, at the end of the course students are given the opportunity to develop a small final group project.

Attendance is highly recommended for this course, both to ensure a full understanding of the theoretical topics covered in the syllabus and strengthen the skills acquired for recognizing optimization models, applying them to economic and financial problems, implementing and solving them using specific optimization software, and critically discussing the solutions.

Student assessment consists of an oral examination including questions covering all topics of the course syllabus, addressing both the theoretical aspects and the real-world applications of optimization models for decision problems in economics and finance. Questions may concern both theoretical and methodological aspects and generally require the student to present a mathematical model and discuss its applications. Questions may also focus on definitions, concepts, properties, and theorems introduced in the more theoretical part of the course, or may require the practical formulation of models for specific problems, or the solution of small numerical examples, possibly using the geometric method.
Different questions are aimed at assessing the achievement of the different levels of knowledge established in Bloom’s Taxonomy, namely Remember–Understand–Apply–Analyze–Evaluate (see the section “Example Questions”).
To pass the exam, students must obtain a mark of at least 18/30. The minimum passing mark is awarded to students who demonstrate that they remember the course topics and understand their meaning.
To obtain the highest mark (30/30 cum laude), students must demonstrate an excellent command of all topics covered during the course, being able to describe and explain them, including the relevant mathematical proofs, as well as to analyze and logically and coherently relate them, consistently using a correct and rigorous formal language.

Question L1: Formally describe the Markowitz optimization model for the selection of the minimum-variance portfolio among those with an expected return at least equal to a given target value.

Questions L2–3: State the definition of an efficient portfolio within the mean–variance framework and illustrate, also through graphical analysis, the procedure for the empirical construction of the Efficient Frontier for a market with n risky assets.

Questions L4–5: Compare the optimization approach of Markowitz’s mean–variance models with that of the alternative Young’s MaxMin model, explaining the differences and discussing the advantages and disadvantages of adopting each model.

Question L1 is intended to assess the achievement of Level 1 of Bloom’s Taxonomy (“Remember”).
Questions L2–3 are intended to assess the achievement of Levels 2 and 3 of Bloom’s Taxonomy (“Understand–Apply”).
Questions L4–5 are intended to assess the achievement of Levels 4 and 5 of Bloom’s Taxonomy (“Analyze–Evaluate”).

• PART 1. Optimization models and techniques
  [24 hours]: Optimization and Mathematical
  Programming – Linear Programming (LP) – Geometrical interpretation of a Linear
  Program – The Simplex Method – Essentials of Duality in Linear Programming. Integer
  and Mixed Integer Linear Programs (ILP, MILP) – Essentials in solution
  algorithms for ILP and MILP – Essentials of multi-objective optimization.
  Essentials of Graphs and Networks and Network flow optimization models.
  PART 2. Applications in finance [16 hours]: Asset Liability models – Capital Budgeting – Portfolio Selection
  models: risk-return models, models based on stochastic dominance, index tracking
  and risk diversification models.
  PART 3. Computational Finance [8 hours]: Introduction to the optimization tools in Excel (and in Matlab) –
  Financial data preparation and processing – Practical solution of optimization
  models for financial problems and portfolio selection (analysis of the results,
  model discussion, post optimization sensitivity analysis).
  • Books: [a] G. Cornuéjols, J. Peña, R. Tütüncü,
    
    Optimization Methods in Finance, Cambridge University Press, 2018. [b] P.
    
    Rardin, Optimization in Operations Research, Upper Saddle River, Prentice-Hall,
    
    1998 (2017).
    
    
    
    [c] F. Cesarone, Computational Finance
    
    MATLAB oriented modelling, Giappichelli Editore, 2020
    
    
    
    [d] E. J. Elton,
    
    M. J. Gruber, Modern Portfolio Theory and investment analysis, John Wiley and
    
    Sons, 1995 (2013). [e] G.L. Thompson,  S.
    
    Thore, Computational Economics, The Scientific Press  1992.
    
    
    
    Teacher handouts 


• Week1 Introduction to optimization mathematical modelling: formal approach for problem modelling and solution –  Introductory examples – Basic notions and definitions – Practical exercises.
  • Books: Teacher handouts and textbooks and [a] Chapter 2 and [b] Chapter 1


• Week2 Linear Programming financial applications examples: Fund allocation problem – Bond allocation – LP scalar and matrix form – LP equivalent transformations and reference forms – General classes of LP models: Resource allocation – Blending models.
  • Books: Teacher handouts and textbook [a] Chapter 2 and [b] Chapter 4


• Week3 Geometrical interpretation of a Linear Program and solution: Feasible region shape and optimal points characterization – Practical exercises. Supplementary Lab activities: Introduction to the Microsoft Excel Solver tool – Solution of a LP – Exercises.
  • Books: Teacher handouts and textbook [a] Chapter 2 and [b] Chapter 2


• Week4 General classes of LP models: Equivalent systems – Standard Form (SF) and LP equivalent transformation into SF. Multiperiod Models: Short-term financing problems – Cash-flow matching and BOND Dedicated portfolio selection – Practical exercises..
  • Books: Teacher handouts and textbook [a] Chapter 2 and 3  and textbook [b] Chapter 5


• Week5 Polyedra and vertices: LP Feasible region – Basic LP Theorems – description of the Simplex Algorithm and Interior point methods search – Examples. Supplementary Lab activities: Practical solution of exercises with Excel.
  • Books: Teacher handouts, Textbook [c], Chapter 1


• Week6 Duality in Linear Programming: Primal-Dual Canonical pair – Dual model construction – Examples and special cases – Duality theorems – Practical exercises (computing the Dual of an LP).
  • Books: Teacher handouts and textbook [a] Chapter 2 and [b] Chapter 6


• Week7 LP Duality and Sensitivity analysis: Shadow prices – (Reduced costs) – Example on the use of  sensitivity analysis on the short-term cash flow model LP presented in Lect. 6 Supplementary Lab activities: Solution of the exercises assigned in the previous lessons. Exercises with Excel Sensitivity Report.
  • Books: Teacher handouts and textbook [a] Chapter 3, Textbook [c], Chapters 1 and 3


• Week8 Integer and Mixed Integer Linear Programming: Knapsack Model and Capital Budgeting Problems – Project Financing Problems. Logic decision variables and constraints –  Modeling fixed costs with 0/1 variables – Examples.   Supplementary Lab activities: Practical solution of model with integer and mixed integer variables. 
  • Books: Teacher handouts textbooks [a] Chapter 8 and [b] Chapter 11


• Week9 Network Optimization models: definitions and basic notions –The Transportation problem and its applications – Minimum Cost Flow Model (MCF) and its applications.Practical Applications: Financial Networks – Generalized MCF and application to a ALM problem. Non linear and Quadratic models optimization: definition and matrix form – particular QPs – Global and local optima – Intuition of optimization procedures  for LP and NLP in the continuous feasible set –  Convex sets, convex and concave functions– Quadratic case. 
  • Books: Teacher handouts textbook [a] Chapter 5 and  [b] Chapter 10[e] Chapter 21 (Cash management problems)


• Week10 Portfolio selection in the risk-return approach: Markowitz’s Modern Portfolio Theory for diversifying risk – Modeling uncertainty for decisions in the risky assets market – Examples –  Basic assumptions in portfolio theory, linear prices and mean-variance approach – Efficient Frontier in a market with n assets. Practical Applications: The Fama’s experiment and illustration of Realistic Examples. 
  • Books: Teacher handouts and textbook [a] Chapter 6, Textbook [d] Chapters 4 and 5


• Week11 Portfolio selection in the risk-return approach: Mathematical Programming for Portfolio selection – Markowitz mean-variance models – empirical construction of the EF for a market of n risky assets – Variants of the Markowitz models with side constraints for modelling requirements on portfolio dividends, upper and lower bounds on the individual positions, cardinality, leverage and turnover constraints, transaction costs,  buy-in thresholds and roundlot transactions – One Fund Separation Theorem. Alternative models in the risk-return approach – MaxMin (expected return) models, Expected Portfolio Return Model (EPRM), MAD model. 
  • Books: Textbook [a] Chapter 6, Textbook [d] Chapter 6 , Textbook [c] Chapter 4[1] Mitra et al. (2003). A Review of Portfolio Planning: Models and Systems, Brunel University, invited chapter in: Advances in Portfolio Construction and Implementation, Satchell, S.E. and Scowcroft, A.E., (Eds.), Butterworth and Heinemann, Oxford. [2] Zenios, S.A. (2002). Practical Financial Optimization: Decision making for financial engineers, Manuscript, HERMES Centre on Computational Finance and Economics, University of Cyprus, Nicosia, CY.


• Week12 Market with a non-risky asset  – The Efficient Line and the Sharpe Index – The Sharpe index maximization problem. Practical Applications: Presentation and discussion of applications in real markets. Supplementary Lab activities: Empirical construction of the EF and of the EL. Advanced Portfolio optimization models: Index Tracking and Risk Diversification optimization models
  • Books: Textbook [a] Chapter 6 Index Tracking [1] Bruni et al., 2015 – Renato Bruni, Francesco Cesarone, Andrea Scozzari, Fabio Tardella (2015). A linear risk-return model for enhanced indexation in portfolio optimization. OR Spectrum, 37, 735–759. [2] Scozzari et al., 2013 – Andrea Scozzari, Fabio Tardella, Sandra Paterlini, Thiemo Krink (2013). Exact and heuristic approaches for the index tracking problem with UCITS constraints. Annals of Operations Research, 205, 235–250 [3] Krink et al., 2009 – Krink, T., Mittnik, S., & Paterlini, S. (2009). Differential evolution and combinatorial search for constrained index tracking. Annals of Operations Research, 172, 153–176. Diversification [1] S. Maillard, T. Roncalli, J. Teiletche (2009). On the properties of equally-weighted risk contributions portfolios. [2] E. Qian (2011). Risk parity and diversification. Journal of Investing, 20, 119. [3] T. Roncalli, 2013 (Book) – Introduction to risk parity and budgeting, CRC Press, 2013. [4] Y. Choueifaty and Y. Coignard (2008). Toward Maximum Diversification. The Journal of Portfolio Management, 35, 40-51.