Ateneo - UPO

First module: financial mathematics.

Second module: linear algebra.

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

M. D’Amico, G. Fusai e G. Longo, Dispensa di Calcolo Finanziario (novembre 2020), Dispensa di Algebra Lineare e sue applicazioni (ottobre 2018), Funzioni di più variabili e Ottimizzazione (novembre 2020); 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.

On module completion, the student should have
1 - a full comprehension of the basic mathematical results regarding the course contents;
2 - being able to choose and use the right tool for solving simple problems related to the course programme;
3 - being able to adapt the instruments to the occurring variants;
4 - being able to adequately and rigorously communicate the logic steps of a problem resolution.

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

Face to face lectures with integrated exercises.
All the informations related to the course as well as supplementary resources
can be found on the course's page on the website www.dir.uniupo.it.

Students with physical disabilities, Learning Disabilities or Special Education Needs can request specific services and tools via the Staff Sviluppo e Coordinamento Carriere e Servizi alle Studentesse e agli Studenti, consulting the University webpage: https://www.uniupo.it/en/services/servicesstudents-physical-or-learning-disabilities Students with disabilities, learning disabilities or special education needs, once they have contacted the University Staff, can refer to the tutor in charge of the course to define the examination modalities, concerning academic aspects.

The exam consists of a written test and an optional oral test. The written test is divided into a laptop based exam (using the SEB system) with numerical questions, constructed in such a way as to verify the knowledge and understanding of the course contents, and with different levels of difficulty (low, medium, high). The questions having a low difficulty level aims to verify a minimum level of understanding of the course topics. The medium-high difficulty questions aim to verifying the ability to autonomously choose the tools suitable for solving complex problems.

The written test can only be taken after passing an online test.
The oral exam is generally optional, but it is compulsory if you want to reach a score higher than 27. The test, in addition to questions aimed at verifying the full knowledge of the concepts presented during the course, tries to verify the ability of autonomous reasoning through the solving problems that differ slightly from those presented during the course.
At the discretion of the course teachers, students may be asked to take a compulsory oral exam.
See the web page of the course at www.dir.uniupo.it

First part – Financial Mathematics. Discounting and capitalization. Usual financial laws (simple and compounded laws), equivalent rates. Annuities. Amortization of debts, Italian and French amortizing schemes. Variable interest rate debts. Financial operations and decisions. Internal interest rate, Italian TAEG. Decisions: net present value (NPV) and adjusted present value (APV). Gordon model. Term structure of interest rates, spot and forward rates. Bond pricing. The boostrapping procedure and recovering the discount curve from bond prices. If time allows: Bond pricing with default risk.

Second part - Linear Algebra. Vectors, matrices and operations. Linear combination, linear independence, rank. Linear systems and linear systems in reduced form. Rouché-Capelli theorem; Gaussian elimination method. Determinant and its properties. Solving linear systems with Cramer method. Application to economics and finance: factor models and risk management, pricing of derivative contracts, performance valuation, bootstrapping and linear systems.

Third part – Optimization. Real functions of several real variables. Gradient and Hessian. Search of extreme points, with and without constraints. Weierstrass theorem. Differential calculus and extreme points search, substitution method. Linear programming with two variables and graphical solution. Applications to business, economics and finance.

At the end of the course student will be able to...
1. Recognize the proper meaning of standard indices of cost/profitability for a financial operation such as NPV, IRR, etc..
2. Identify the proper meaning of terms such as yields, default risk and so on.
3. To be able to adopt the correct procedures for solving linear systems, financial and optimization problems.

APPLYING KNOWLEDGE AND UNDERSTANDING
At the end of the course student will be able to apply the learned methods to compute quantities relevant both in theory and in practice such as: the price of a bullet bond, the internal effective rate of a loan, the probability of default, the extra-performance of a fund, assess the profitability of a project.