Degree course: 
Corso di First cycle degree in ECONOMICS AND MANAGEMENT
Academic year when starting the degree: 
Academic year in which the course will be held: 
Second semester
Standard lectures hours: 

None. However, it is strongly recommended to have acquired the skills of Mathematics I and Mathematics II courses.
The topics of the Statistics course can be a further aid for participation in the lessons.

Final Examination: 

According to University regulations and up to different indications, in the academic year 2020/21 the exam will be carried out remotely. The remote exam will be carried out on a specific University platform (e-learning). A student can decide to carry out the exam in two different methods.
METHOD 1 – Total exam:
At the end of the course, during the exam sessions, tests that cover the entire program of the course will be organized. During the total exam (at a distance), the use of calculators, manuals and notes will be allowed.
The test will be divided into two parts and will be designed to evaluate the calculation skills, the knowledge of terminology, the main statements presented during the course and the analytical skills developed by the student.
Part 1: The first part of the exam will be composed by Quiz (multiple choice) delivered through the e-learning platform. The student will have to answer 6 multiple choice or numerical answer questions in 20 minutes. By correctly answering at least 4 questions, students will be admitted to the second part of the exam. Otherwise the exam will be terminated, and the test will be evaluated as insufficient. There is no penalty for incorrect answers. Each correct answer is worth 2 points.
Part 2: The second part will consist of two exercises, composed by several questions. Each exercise is worth up to a maximum of 10 points.
The student will have 40 minutes to solve the exercises on a sheet of paper with black pen and upload a legible image of the paper to the e-learning platform. Further details on how to create and upload the solutions of the exercises will be made available on the e-learning page of the course.
The exam is passed if the sum of the scores obtained in the two parts is not less than 18 (eighteen), with a minimum of 8 (eight) points in the first part. Scores higher than 30 will be registered as 30 with honors.

MODE 2 - Partial exams:
At the end of the first cycle of lessons, in the week of teaching interruption, and at the end of the course in December, two partial exams will be organized mainly concerning the topics of the part of the course just ended.
The first partial exam will last 40 minutes, will be worth 16 points and consists of 12 short questions in the form of quiz (multiple choice question) concerning calculation skills and knowledge of the main definitions and theoretical concepts addressed in class. Students will be admitted to the second partial exam if the score obtained in the quizzes is greater than or equal to 6.
The second partial exam is divided into two parts. The first part of the exam, worth 6 points, is composed of short questions in the form of quiz (multiple choice question). The student must achieve at least 4 points to be able to access the second. The second part of the exam, evaluated 10 points, is composed of more complex exercises, in which the student is required to use the calculation skills and theoretical results presented in the course in an appropriate way to provide the solution to the proposed questions. There is no minimum score for this part.
The second partial exam is also passed with at least 6 points.
The exam is passed if both partial tests are passed and the sum of the points obtained is no less than 18 (eighteen). Honors are awarded to a sum greater than 30.
Students with DSA: Students with DSA are required to contact the disabled service ( to define the individualized Training Project that must be sent to the course holder within 10 days before the exam session that the student intends to sustain.

Voto Finale

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.

Main topics cover:
• Capitalization and discounting: financial laws, financial laws in one and two variables.
• Standard Financial Contracts: depreciation, leasing and installment plans.
• legal indicators: TAN, APR, ISC.
• Financial valuations of non random investments.
• Term structure of interest rates and fixed-income securities.
• Financial immunization.
• Probability: different approaches to the definition and study.
• random experiments, events, probability, Bayes' Theorem.
• random variables: probability distribution and their moments.
• Expected value and variance of a linear transformation (affine) of a random vector.

1. Probability:

• Introduction to probability. Random experiment. Classical, frequentist, subjectivist approach. Axiomatic approach. Results space. Random events, algebra / σ-event algebra. Probability of an event, probability space. Axioms of probability. Additive / σ-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.

2. 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 r

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.


The course includes 30 hours of 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.
An additional 30 hours will be fulfilled through an individual guided study through the University e-learning platform.
Exercises in preparation for the exams will be provided.

Starting from the summer session of the academic year 2019/2020 the examination procedures and the program will be those described. Students who have attended the course in previous years will have to take the exam with the new methods.