# FINANCIAL MATHEMATICS

SECS-S/06 - 6 CFU - 2° Semester

SALVATORE GRECO

## Learning Objectives

Knowledge and understanding. The course provides the theoretical principles underlying the time value of money under certainty (rates and their structure, rules of capitalization, depreciation, capital formation, evaluation of loans, bonds, investment analysis). It also provides tools to manage interest rate risk (duration and convexity). Moreover, basic notions about the theory of functions with many variables are supplied with a specific focus on their applications in economics, management and finance. Alongside the theoretical knowledge, properly formalized, the course is intended to transfer professional skills. Indeed, the topics covered are explained paying attention to the operational point of view, in order to provide the knowledge needed to apply the methods and the techniques studied to real world problems (know how to evaluate, compare, making decisions). To achieve these goals, appropriate equipment and teaching supports are used, such as multimedia presentation, database accessing, spreadsheets. The whole training of the discipline also aims at developing the inductive-deductive logical process of the students’ learning. The final examination (structured as written and oral tests) is not the ultimate goal. During the whole course there is a continuous checking of the comprehension and the real acquisition of the taught knowledge. There is an active participation of the students to the teaching process.
Applying knowledge and understanding. Particular attention is paid to stimulate the professional skills of the potential graduates. To this end, lecturers use a didactic methodology oriented towards the acquisition of operational capacities (know-how) concerning the analytical tools and the theoretical concepts provided during the course. Real world cases are often submitted to the students. The final examination must verify the effective acquisition of these skills.
Making judgments. The development of a critical understanding of the topics provided by the course is a major educational objective. The learning skillfulness must be accompanied by a thorough capacity of evaluating, assessing and solving a problem using the most appropriate methods and techniques, whereas the student is asked to check the proper limits. During the whole course, the interaction with all the students is fundamental, and is pursued in a constructive way with the aim of stimulating the ultimate understanding of the information needed to set up, analyze and solve the fixed problems, avoiding a sterile mnemonic preparation. The students are trained via the most appropriate economic and financial sources (academic publications, databases, Internet sites, etc.). The students will also learn to analyze and evaluate the reliability and meaningfulness of such information and data, to use them appropriately by dealing with real world problems.
Communication skills. It is not enough to be able to apply correct methods and techniques to deal with the problem at hand. It is also necessary to justify them, revealing the underlying assumptions on which the analysis is based. To this end, besides the appropriate theoretical knowledge, in view of its practical implementation, it is important to learn using computational tools and multimedia technology. The course is then designed in order to develop these skills, by ensuring the active participation of the students. They are asked to illustrate their understanding via written notes, and to prepare presentations individually and in groups. These are discussed in the classroom. The final exam has the additional aim of verifying the communication skill developed by the students during the course.
Learning skills. The students are asked to improve their method of study, in view of a more effective learning of the arguments in the program. The checking of the actual acquisition of theoretical and operational knowledge, necessary for entering the job’s world, takes place during the entire course. The lecturer possibly reviews his method of teaching. The learning process is then constantly monitored and improved, avoiding a negligible approach.

## Course Structure

The course is divided in thre main parts. In the first part, the basic concepts of classical financial mathematics related to financial conventions, annuities, amortizations, founding capital are presented. In the second part, some specific aspects of modern financial mathematics related to valuation of financial and real investments are treated. In the third part, basic notions on the theory of real valued functions of real variables are supplied with a specific focus on their applications in economics, management and finance.

## Detailed Course Content

MODULE # 1 (3 CFU) Financial conventions, annuities, amortizations, founding capital

Learning goals: Providing both the theory and practice of elementary financial calculus under certainty. As a byproduct, this helps to develop professional skills.

Topic description: The financial function and its properties. Financial convention: simple, commercial and compound; mixed cases; rational vs commercial discount. Equivalent interest rates, nominal interest rates, instantaneous convention. Annuities and their classification: general discrete, periodic, constant, fractional, continuous, perpetual. Annuities in compound convention: periodic arithmetic and geometric progression payment; perpetuities. Inverse problems. Unshared loan and amortization: general properties. Compound convention in amortization: Single settlement repayment; multiple settlement repayments: general weak amortization installments; several interest repayments and single repayment of the principal (general and periodic); several interest repayments and single repayment of the principal with collateral funding of the principal: general case. American amortization. Italian amortization. French amortization. German amortization. Cession’s value of rights concerning a loan’s amortization. Capital accumulation: discrete case.

MODULE # 2 (3 CFU) Valuation of financial and real investments
Learning goals: Providing the theory and the main techniques for evaluating both financial and real investments. Explaining the concept of interest rate risk and the corresponding techniques of immunization.

Topic description: Loan evaluation and general investment evaluation. Bare ownership and usufruct. Investments in real markets under certainty. Some useful criteria of investment evaluation: Net Present Value (NPV); Internal Rate of Return (IRR); Payback period. Comparison among criteria. Shared loan amortization: basic concepts. Constant amortization installments, constant reimbursement price. Effective rate for the issuer; cession’s value of the credit; effective rate for the holder. Cession’s value of a bond. Bond’s market: prices vs rates/yields. Zero coupon bonds. Fixed coupon bonds. The structure of the market. Forward rates and spot rates. Immunization: basic principles. Interest rate risk. Theorems of immunization: parallel and nonparallel shifts. Time indexes: arithmetic mean maturity; duration and modified duration. Convexity.

MODULE # 3 (3 CFU) Real valued functions of real variables
Learning goals: Providing the basic notions of the theory of real valued functions of real variables and the related main techniques and methodologies useful for their applications in economics, management and finance.

Topic description: Definition of basic concepts. limits and continuity. Partial derivatives and gradient. Total differential. Homogenous functions and implicit functions. Unconstrained and constrained optimization. Applications in economics, managementand finance..

## Textbook Information

R. L. D’Ecclesia, L. Gardini, Appunti di Matematica Finanziaria I, VIII edizione, Giappichelli, Torino, 2019

Open in PDF format Versione in italiano