FINANCIAL AND ACTUARIAL MATHEMATICS

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.

The course is divided in three 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.

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, management and finance..

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

S. Corrente, S. Greco, B. Matarazzo, S. Milici, Matematica Generale, II edizione, Giappichelli, Torino, 2021

The examination consists of an oral test in which adequate knowledge and mastery of all the topics in the syllabus is ascertained and the student's ability to use and appropriately apply the basic concepts, tools and fundamental results proposed in the syllabus is verified on the basis of the performance and commentary of one or more exercises. The grade will be awarded on the basis of the level of preparation demonstrated by the student, it being understood that passing the examination requires the attainment of a minimum threshold of knowledge of the topics covered in the syllabus. The learning assessment may also be conducted electronically, should the conditions require it.

What are interest, discount, future and present value?

What are the interest rate and the discount rate and what is their functional relationship?

What is a capitalisation law?

What are simple, compound and commercial capitalisation regimes?

Can you compare simple, compound and commercial capitalisation regimes?

When are two rates said to be equivalent?

What are interest force and discount force?

What is decomposability?

What is the necessary and sufficient condition for a capitalisation law to be decomposable?

How are the present value and the amount of a deferred annuity of n constant instalments determined?

What is the present value and the final value of a deferred annuity of n instalments in arithmetic progression?

What is the present value and the final value of a deferred annuity of n instalments in geometric progression?

What are the differences between French, Italian, American and German amortisation?

What are bare ownership and usufruct?

What is, how is it derived and how is Makeham's formula used?

What are internal rate of return and net present value?

What are spot rates and forward rates and what is the relationship between them?

What are duration and convexity?

Can you state the Fisher-Weil theorem?

Can you state the Redington Theorem?

What is a contour line?

When does a function converge for x that tends to x_0?

When is a function said to be continuous?

Can you state the Weirstrass theorem?

What is the partial derivative?

What is a partial derivative of order m?

Can you tell me Schwarz's Theorem?

What is the total differential?

What are the directional derivative and the radial derivative?

What is the total derivative of a compound function?

What is a homogeneous function of degree r?

Can you tell me Euler's theorem?

What is an implicit function?

Can you tell me the Dini theorem?

What is Taylor's formula?

When is a function convex? 

What is a quadratic form?

When is a quadratic form said to be sign defined?

What is a local minimum (or maximum)? 

Can you tell me  necessary conditions for the existence of a local minimum (or maximum)?

Can you give me sufficient conditions for the existence of a local minimum (or maximum)?

Can you tell me Lagrange's theorem?

Can you state the Kuhn-Tucker theorem?