# MATHEMATICS FOR ECONOMICS AND FINANCE

- Overview
- Assessment methods
- Learning objectives
- Contents
- Full programme
- Bibliography
- Teaching methods
- Contacts/Info

To be admitted at the exam, students must first pass Matematica.

Written exam.

Students can take the exam alternatively with:

One general test.

At the end of the course, during the examination sessions, 90 minutes examinations will be held on the entire course program. To pass the exam, the student must achieve a grade not lower than 18/30 (eighteen). Totals above 30 give the right to honor mark.

The test consists of two parts.

The first is composed of 10 short questions (open or closed answers), also of a theoretical nature. By answering less than 7 questions correctly, the test is not passed. By correctly answering at least 7 questions, the outcome of the second part will be evaluated. By correctly answering more than 7 questions, points will be added to the outcome of the second part:

8 correct questions + 1 point

9 correct questions + 2 points

10 correct questions + 2 points

The second part consists of 3 open-ended exercises, each of which is assessed up to 10 points.

Two partial tests.

At the end of the first cycle of lessons, and at the end of the course, two partial tests will be organized concerning the topics of the part of the course that has just been completed (the first will focus on "calculation of probabilities", the second on "financial calculation"). Each test, evaluated in sixteenths, be considered passed with a score not lower than 7 (seven). The exam is passed if the sum of the scores of the two tests is not less than 18/30 (eighteen). Totals over 30 give the right to praise.

The first partial test will last 40 minutes, is evaluated 16 points and consists of 10/12 short questions in the form of quizzes (multiple choice questions) designed to evaluate the calculation skills and knowledge of the main definitions and theoretical concepts addressed in the course.

The second partial test is divided in two parts and will last 60 minutes. The first part is composed of 5 short questions (open or closed answers), also of a theoretical nature. Responding correctly to less than 3 questions, the test is not passed. By correctly answering at least 3 questions, the outcome of the second part will be evaluated. By correctly answering more than 3 questions, points will be added to the outcome of the second part:

4 correct questions + 1 point

5 correct questions + 2 points

The second part consists of 2 open-ended exercises, each of which is evaluated up to 7 points.

Students with LD are required to contact the disabilitiesâ€™support office (servizio.disabili@uninsubria.it) to define the individualized training project to be sent to the course owner within 10 days of each exam session to be taken.

The course gives the fundamental concepts of probability theory and its applications to economics and finance. Moreover, it offers the basic notions of mathematical finance. At the end of the course students should be able to:

a) Construct common financial contracts, such as leasing contracts, consumer credit and loans;

b) Compute some legal indicators linked to financial contracts, such as TAN, APR, ISC;

c) Evaluate the convenience of some financial transactions according to the most accredited criteria that allow the maximization of profit in a given time horizon;

d) Critically comment on criteria inconsistent with the goal of maximizing profit as set time horizon;

e) Compute prices and yields of main fixed income securities, as well as the Duration and the underlying term structure;

f) Apply basic probability to managerial and financial problems

The course will be held with lectures in which we will discuss the theoretical arguments and the methods of solution of the exercises. During classes, exercises will be proposed both for self assessment and class work.

Main topics cover:

â€¢ Probability: different approaches to the definition and study. (3h)

â€¢ random experiments, events, probability, Bayes' Theorem. (10h)

â€¢ Random numbers: probability distribution and their moments (10h)

â€¢ Expected value and variance of a linear (affine) transformation of a random vector (7h)

â€¢ Capitalization and discounting: financial laws, financial laws in one and two variables. (7h)

â€¢ Standard Financial Contracts: depreciation, leasing and installment plans.(6h)

â€¢ legal indicators: TAN, APR, ISC.(4h)

â€¢ Financial valuations of non random investments.(5h)

â€¢ Term structure of interest rates and fixed-income securities.(5h)

â€¢ Financial immunization.(3h)

Probability:

Introduction to probability. Random experiment. Classical, frequentist, subjectivist approach. Axiomatic approach. Results space. Random events, algebra / Ïƒ-event algebra. Probability of an event, probability space. Axioms of probability. Additive / Ïƒ-additive; conditional probability. Stochastically independent, positively / negatively related events. Bayes theorem and its applications.

Random numbers. Borel-measurable functions. Random numbers, distribution function. Discrete random numbers, probability function. Examples of discrete distributions: uniform, binomial, Poisson. Continuous random numbers, probability density function. Examples of continuous distributions: uniform, exponential, normal. Expected value of a random number. Variance and standard deviation of a random number. Expected value of a function of a random number. Moments of a random number, reinterpretation of expected value and variance in terms of moments. Expected value, variance and standard deviation of an affine linear function of a random number.

Discrete random vectors. Random vectors. Two-dimensional random vectors: distribution function, stochastically independent random numbers. Discrete two-dimensional random vectors: probability function (joint and marginal). Vector of expected values of a random vector. Expected value of an affine linear function of a random vector. Covariance between two components of a random vector. Linear correlation between components; linear correlation coefficient of Bravais. Variance-covariance matrix of a random vector. Variance of an affine linear function of a random vector.

Financial calculus:

Financial laws. Capitalization, discount (or discounting). Initial capital, amount, interest, capitalization factor (or amount); nominal value at maturity, present value (or discounted value), discount, discount factor. Interest rate, discount rate. Financial laws, financial regimes. Conjugated factors. Financial laws of a variable. Ordinary schemes: simple capitalization, compound capitalization, capitalization at simple interest rates in advance; rational discount (or simple discount), compound discount, trade discount. Equivalent rates in simple and simple capitalization in advance. Equivalent rates in compound capitalization; convertible nominal annual rate, effective annual rate. Generality levels for laws of a variable. Follow-up factor. Financial laws of two variables. Generality levels by laws of two variables. Instant intensity of interest for laws of a variable; the case of compound capitalization. Separable by laws of a variable. Instant intensity of interest for laws of two variables. Separable by laws of two variables. Separability by product, Cantelli theorem. Application: the actuarial capitalization factor.

Cash Flows. Financial transactions, cash flows. Current value and amount of a cash flow. Annuities, investments, financing. Periodic rents in constant installments, in compound capitalization (temporary / perpetual, postponed / advanced); the symbols. DCF, VAN (or NPV), internal rates of a financial transaction. DCF chart for investments and financing. TIR (or IRR) of an investment, actual cost of a loan.

Depreciation. Amortization of a debt. Maturities, installments, capital shares, interest quotas, residual debt, extinguished debt, total interest. Amortization plan. Closing conditions: elementary, initial, final. Elementary approach, financial approach. The case of compound capitalization. Contract rate. Formulas for interest rates, recursive formulas for residual debts. Change of conditions. Italian amortization, French amortization. Applications: consumer credit, leasing. Rate calculation problems; time profile of payments. Rate calculation problems; TAN, APR.

Term structure. The term structure of prices. Spot (spot) prices of zero-coupon bonds, forward (forward) prices of zero-coupon bonds per unit. Arbitrages without risk. Impossibility of arbitrage without risk

Attendance is not mandatory, though highly recommended. Nevertheless, it is compulsory to study course material through textbook. The adopted texts are:

1.E. CASTAGNOLI, L. PECCATI, Matematica in azienda 1 (Calcolo finanziario con applicazioni), Milano, Egea, 2010.

2.E. CASTAGNOLI, M. CIGOLA, L. PECCATI, Probability. A Brief Introduction, Milano, Egea, 2009.

Recommended reading for further information:

Sheldon Ross, ProbabilitaÌ€ e statistica per lâ€™ingegneria e le scienza, Apogeo Education, Maggioli Editore (terza edizione).

The course will take place with lectures in which the theoretical topics and the methods of solution of the exercises will be discussed. During the lessons exercises can be assigned both for individual study and to solve in the classroom.

Exercisesâ€™ sessions will be held during the semester.

On the e-learning page of the course (class H-Z) https://elearning.uninsubria.it/course/view.php?id=10855 general information on the course, the materials of the lessons, the exams of the previous years, the exercises and any additional exercises will be made available. Students are invited to consult it periodically. It is possible to ask for clarification at the end of each lecture or during the office hour. Students can also fix an appointment by sending an email at the address: matteo.rocca@uninsubria.it (class A-G); elisa.mastrogiacomo@uninsubria.it e enrico.moretto@uninsubria.it (class H-Z)