QUANTITATIVE METHODS
To be admitted at the exam, students must first pass Matematica I.
The exam is written for both modules. The exam may be taken in two different ways:
A general exam, whose format is the same for both modules: MOD. 2 – Mathematics II and MOD. 1 – Statistics.
At the end of the course, during the exam sessions, written exams with a total duration of 90 minutes covering the entire course syllabus will be organized.
To pass the exam, students must obtain a grade of at least 18/30 (eighteen). Scores above 30/30 entitle students to honours.
The exam consists of two parts:
The first part, lasting 20 minutes, consists of 10 short questions (open-ended or multiple-choice), including theoretical questions. With fewer than 6 correct answers, the exam is failed. With at least 6 correct answers, the outcome of the second part is evaluated. With more than 6 correct answers, bonus points are added to the score of the second part: 8 correct answers → +1 point; 9 correct answers → +2 points; 10 correct answers → +3 points.
Only students who pass the first part are admitted to the second part, which lasts 70 minutes and consists of 3 open-ended exercises, each graded up to 10 points.
Two midterm exams.
At the end of the first cycle of lectures and at the end of the modul, two midterm exams will be organized, covering the topics just completed of each module. Each midterm exam, graded out of 16 points, is considered passed with a score of at least 7 (seven). The exam is passed if the sum of the scores of the two midterms is at least 18/30 (eighteen). Scores above 30/30 entitle students to honours.
For MOD. 2 – Mathematics II, the first midterm exam has a duration of 40 minutes, is graded out of 16 points, and consists of 10–12 short quiz questions (multiple-choice).
Students are admitted to the second midterm exam if the score obtained in the first midterm is greater than or equal to 7.
The second midterm exam has a total duration of 60 minutes and is divided into two parts.
The first part consists of 6 short questions: with fewer than 3 correct answers the exam is failed; with at least 3 correct answers students are admitted to the second part. More than 3 correct answers yield bonus points (+1 point with 4 correct answers, +2 with 5, +3 with 6).
The second part consists of 2 open-ended exercises, each graded up to 7 points.
Only for MOD. 2 – Mathematics II, during the course four surprise mini-tests will be administered.
Each mini-test consists of 5 questions; each correct answer is worth 0.1 points. The total score of the mini-tests constitutes a bonus added to the final exam grade. A student who answers all questions correctly in all four mini-tests may obtain a maximum of 2 bonus points. The bonus is valid only if the exam is passed during the 2025/2026 academic year.
For MOD. 1 – Statistics, the first midterm exam has a total duration of 60 minutes and is divided into two parts.
The first part consists of 6 short questions (open-ended or multiple-choice): with fewer than 3 correct answers the exam is failed; with at least 3 correct answers students are admitted to the second part. More than 3 correct answers yield bonus points (+1 point with 4 correct answers, +2 with 5, +3 with 6).
The second part consists of 2 open-ended exercises, each graded up to 7 points.
The second midterm exam has the same format as the first and is open to students who obtain at least 7 in the first midterm.
Students with DSA are required to contact the Disability Service (servizio.disabili@uninsubria.it) to define the Individual Educational Plan, to be sent to the course instructor at least 10 days before each exam session they intend to take.
This course is divided in two modules: MOD. 1 STATISTICS and MOD. 2 MATHEMATICS II.
MOD. 1 STATISTICS aims to provide students with fundamental knowledge of statistical data analysis methodologies in the economic field.
Specifically, the course seeks to equip students with both theoretical and applied tools necessary for conducting a rigorous statistical analysis of a dataset. Students will learn how to extract useful information while also assessing its reliability.
EXPECTED LEARNING OUTCOMES
By the end of the course, students will be able to:
Describe and understand the main concepts of descriptive and inferential statistics;
Summarize a dataset appropriately;
Apply statistical tools to economic and business-related problems;
Construct simple statistical models, such as regression models, to study relationships between different variables of interest;
Estimate and test hypotheses about unknown population parameters based on sample data;
Critically interpret and comment on key results obtained.
MOD. 2 MATHEMATICS II gives the basic notions of mathematical finance. Moreover, it offers the fundamental concepts of probability theory and its applications to economics and 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 (even under uncertainty) 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 program for MOD. 2 MATEMATICA II is:
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 and its consequences. Relationship between forward prices and spot prices, link with the separability of the financial law expressed by the market. Non-arbitrage prices of securities with coupons. The term structure of rates. Spot rates, forward rates. The case of flat term structure.
Fixed income securities. Securities without coupons (zero-coupon bonds): gross compound return; simple gross return, yield from sale before maturity, links with non-divisibility; net simple return, with taxes on issue or redemption. Securities with constant coupon. Issuing conditions: nominal value, issuing price (or clean price), redemption value; issuing discount, redemption premium; coupons, annual coupon rate. Conditions of purchase: course tel quel, course ex-coupon, clean course. Price dependence on the term structure of market rates. Yield to maturity, Current yield. Duration of a security. Financial immunization. Duration of a portfolio. Volatility of the price of a security, compared to changes in the market rate. Modified duration. Approximate price changes using the Duration.
Financial assessments. General notions. The opportunity cost of equity. Assessing the value of cash flows through the NPV; underlying assumptions and its equivalence with the maximization of wealth at a future time. the NPV as a transfer price threshold. The case of an investment: using the IRR. The case of a loans: assessing through the actual cost. Critics to the internal rate criterion. Limits of the NPV and its generalizations: generalized NPV (or GNPV), APV (or NPV on equity), GAPV (or generalized