Ateneo - UPO

First module: financial mathematics.

Second module: linear algebra.

Third module: real functions of multiple real variables and their optimization.

G. Fusai, M. D’Amico, A. Gambaro e G. Longo, Dispense di Calcolo Finanziario, Algebra Lineare e Ottimizzazione, year 2026, available on the DIR repository. Addtional useful references F. Privileggi, Matematica per l’Economia. Gruppo editoriale Esselibri – Simone, 2008. S. Margarita - E. Salinelli, MultiMath: Matematica Multimediale per l’Università, Springer, 2004.

Upon successful completion of the course, students will be able to use the fundamental tools of linear algebra, differential calculus, and optimization to analyse and solve basic quantitative problems in economics, finance, and management.

In particular, students will be able to perform the main operations involving matrices and vectors, calculate determinants and matrix inverses, and solve systems of linear equations, interpreting their solutions within the context of the problem under consideration.

Students will also be able to analyse functions of two or more variables using the tools of differential calculus, calculate partial derivatives, and identify stationary points. They will be able to determine and classify maximum and minimum points and address basic unconstrained and constrained optimization problems.

Finally, students will understand the main interest accumulation regimes and the fundamental concepts of the time value of money, accumulation, discounting, interest rates, and equivalent rates. They will be able to use the main financial appraisal criteria and compare investment projects characterized by deterministic cash flows, applying appropriate and consistent decision criteria.

The exam Metodi Matematici 1 has to be passed before taking Metodi Matematici 2.

Face to face lectures with integrated exercises.

The course page available on the teaching platform www.dir.uniupo.it provides all relevant information concerning the organization of the course and the examination procedures. Additional teaching materials, exercises, and examination papers from previous academic years, including both questions and solutions, are also made available through the platform.Students with disabilities, Specific Learning Disorders, SLDs, or Special Educational Needs, SENs, may request dedicated services, support, and specific arrangements by contacting the University Staff responsible for Student Careers and Student Services and consulting the dedicated page on the University website:https://uniupo.it/it/servizi/servizi-studentidisabili-e-dsaAfter contacting the relevant University Staff, students with disabilities, SLDs, or SENs may contact the course instructor to discuss the implementation of the approved measures, including any specific arrangements relating to the examination.

Examination ProceduresPreliminary Technical AssessmentTo be admitted to each examination, students must first pass a preliminary Technical Assessment consisting of three questions.The Technical Assessment is completed independently by the student, and all questions must be answered correctly. Successful completion of the assessment is a mandatory requirement for admission to the subsequent examination.Each examination has its own preliminary Technical Assessment, which must be completed and passed in accordance with the procedures and deadlines communicated by the course instructor.Written ExaminationThe written examination is administered through the Esami DIR platform using Safe Exam Browser.The full examination lasts 80 minutes and consists of 21 questions covering the three main areas of the course, Financial Mathematics, FM, Linear Algebra, LA, and Optimization, OPT.The final score is calculated as the sum of the four highest scores obtained in the Financial Mathematics questions, the two highest scores obtained in the Linear Algebra questions, and the three highest scores obtained in the Optimization questions.The partial examinations follow a similar format. The duration, number of questions, and specific assessment criteria will be communicated separately for each examination.Compulsory Confirmatory Oral ExaminationStudents who achieve a passing grade in the written examination are required to take a confirmatory oral examination immediately after completing the written assessment.The oral examination will cover the questions included in the Preliminary Technical Assessment and the written examination just completed. Students must be able to explain and justify the procedures followed and the answers provided.Failure to attend the confirmatory oral examination or inability to provide an adequate justification for the answers given may result in the written examination being declared invalid.Optional Oral ExaminationStudents who have achieved a passing grade in the written examination and successfully completed the compulsory confirmatory oral examination may take an additional oral examination to improve their final grade. Without the optional oral examination, the maximum grade that can be awarded is 26 out of 30.The optional oral examination will cover the theoretical topics included in the entire course syllabus and will not involve numerical exercises.In addition to answering questions posed by the instructor, students may select one topic from the course syllabus to present and discuss in greater depth.

First Module, Financial MathematicsElementary financial transactions: cash amounts, payment dates, principal, accumulated value, interest, interest rates, present value, nominal value, discount, and discount rates.Financial laws involving one time variable: accumulation and discount factors. Standard interest regimes, simple interest and compound interest. Equivalent interest rates.Financial annuities: classification of annuities and calculation of their present and accumulated values.Amortization of loans: structure of amortization schedules, Italian and French amortization methods. Variable rate mortgages.Evaluation and comparison of financial transactions: internal rate of return, IRR, annual percentage rate of charge, APRC, net present value, NPV, and the Adjusted Present Value, APV, approach. Valuation of leasing contracts.Applications to business valuation: the Gordon Growth Model and fundamental principles of company valuation.Term structure of interest rates: spot and forward interest rates. Valuation of default free bonds.Second Module, Linear AlgebraVectors and matrices. Main matrix operations, addition, multiplication, transposition, and inversion, and their properties.Linear combinations. Linear dependence and independence. Matrix rank and elementary row operations.Systems of linear equations: conditions for the existence and uniqueness of solutions, the Rouché, Capelli theorem, and solution by Gaussian elimination.Economic and financial applications of linear algebra and systems of linear equations, with particular reference to financial risk management and performance measurement.Third Module, OptimizationReal valued functions of several variables: domains, graphical representations, and level curves. An introduction to the concepts of limits and continuity.Local and global maximum and minimum points. Unconstrained and constrained optimization problems. The Weierstrass theorem.Differential calculus for functions of several variables: partial derivatives, gradients, second order derivatives, and the Hessian matrix.Unconstrained optimization problems: identification of stationary points, first order necessary conditions, and second order sufficient conditions for the classification of maximum and minimum points.Constrained optimization problems with equality constraints: the substitution method and the level curve method.Introduction to linear programming: formulation of objective functions and constraints, graphical representation of the feasible region, and identification of optimal solutions.Applications to business, economic, and financial problems, with particular reference to resource allocation and portfolio selection.

Intended Learning Outcomes

Upon successful completion of the course, students will be able to:
1. use the fundamental tools of linear algebra, perform the main operations involving vectors and matrices, and solve systems of linear equations, understanding how these methods can be applied to the modelling and solution of economic and financial problems;
2. analyse functions of two or more variables using the tools of differential calculus, calculate partial derivatives, identify and classify maximum and minimum points, and apply the main methods of unconstrained and constrained optimization to economic, financial, and business problems;
3. understand and apply the main interest accumulation regimes, correctly use the fundamental quantities and indicators of financial mathematics, and evaluate and compare financial transactions and investment projects characterized by deterministic cash flows, applying appropriate and consistent decision criteria.