Scientific Disciplinary Sector (SSD)
SECS-S/06 - MATHEMATICAL METHODS OF ECONOMICS, FINANCE AND ACTUARIAL SCIENCES
Primo semestre (lauree) dal Sep 19, 2022 al Jan 13, 2023.
The course aims to provide knowledge of classic mathematical models for the analysis and evaluation of the main financial operations, financing contracts and economic-financial investment projects. The models discussed also include the one for selecting an optimal investment portfolio in terms of return-risk, in the mean-variance approach.
At the end of the lessons, the student must be able to correctly set up a financial evaluation problem, taking into account the right hypotheses to be posed, the correct choice of mathematical expressions in the financial regime adopted and the correct mathematical methodology for solving the problem itself; therefore the teaching aims to develop the capacity for independent reasoning in the analysis and interpretation of concrete assessment and choice problems. In the lessons there will be a continuous alternation between the treatment of the general aspects of the models of financial mathematics, the illustration of examples of a real nature and the carrying out of exercises which equally propose the study of concrete situations. In exercises, especially in computationally heavy cases, scientific calculation software will be used. As the lessons progress, the students will find further learning exercises on the various topics covered in the course's E-learning pages.
Prerequisites and basic notions
The course only requires the knowledge of basic notions of mathematics (polynomial functions, exponential functions and their properties). More advanced notions (derivation, even in several variables, and use of the derivative to determine the maxima and minima of functions) will be recalled before their use.
Basic financial operations: capitalization and actualization. Financial laws: amount, present value, interest, discount, interest rate and discount rate. Regime of simple interest, compound interest, commercial discount. Equivalence between different laws and between interest rates over periods other than the year. Strength of interest. Spot rates, forward rates. Nominal rates, inflation, real rates.
Compound financial operations and their classification. Present and final value of a financial annuity. Net present value (NPV). Case of constant annuities. Internal rate of return (IRR). Consumer credit, Annual nominal rate and annual percentage rate (APR).
Amortization plans and closing conditions. Various types of amortizations. The pre-amortization. A simple model for financial leasing.
Bonds. Valuation of the price of a bond. BOTs and BTPs and their valuation. Estimate of the interest rate term structure. Interest rate risk control / immunization: duration and convexity.
Criteria for choosing between financial projects: Discounted economic return / NPV / IRR / TRM.
Some reminder of mathematical analysis. A model for selecting the financial portfolio. Risk and return. Portfolios of two or more assets. Stochastic dominance. Diversification. Efficient frontier. Capital asset pricing model (CAPM). Systematic risk and diversifiable risk. Risk premium.
The course gives 9 credits, corresponding to 72 hours of lessons. On the various topics classical lessons of a general nature will alternate with hours of an application nature, with the carrying out of exercises, also through reference to the use of software (in particular Excel of MSOffice / Calc of LibreOffice). The lectures are in Italian.
On the teaching web page, students have access to supplementary notes written by the teacher and all the previous exam topics and solutions. In the e-learning pages they may also find the texts and the developments of the exercises carried out weekly in the classroom.
During the semester of lessons and before the exam sessions, a tutoring service is active, held by students of the Master's Degree Courses and coordinated by the teacher.
Learning assessment procedures
The final exam consists of a written test, possibly followed by an oral test only in cases of not serious insufficiency in the written test (marks between 16/30 and 17/30). The test includes 3-4 exercises, each divided into several questions.
Halfway through the course, there will be an intermediate written test on the topics already addressed in the course. The intermediate test, similar in structure to the written test of the ordinary sessions, is designed with the aim of stimulating students to systematically study the subject during the course, in order to improve the learning process.
The intermediate test is optional; participation in the test does not affect the right to refuse the grade obtained and therefore to take the entire exam at the end of the course.
The grade obtained in the intermediate test weighs half of the final grade, which is achieved by passing a similar second final written test. The exam is passed if and only if both partial tests are passed, possibly with the need to take the supplementary oral exam in cases of minor insufficiency.
The topics of the mid-term test will be specified in a specific notice on the e-learning page of the course, well in advance of the test date. The topics covered by the mid-term exam are excluded from the written completion test at the end of the semester, without prejudice to the inevitable correlations between the topics, typical of the discipline. Useful rounds for the completion of the exam are the ones in the session immediately following the end of the course, namely the two rounds of the winter session.
The score of each exercise or part of the exercise is proportionate to the difficulty of the same. The sum of the partial scores of the test is equal to 30 points.
Criteria for the composition of the final grade
The final mark of the written test is given by the sum of the partial scores. Students with sufficient results can always ask to take a supplementary oral test: in this case the mark of the written test is only a starting point, but neither the maintenance nor the increase of the mark itself is guaranteed, nor the passing of the exam. The same is true when the oral exam is compulsory.