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Course 2022-2023 a.y.


Department of Decision Sciences

For the instruction language of the course see class group/s below

Go to class group/s: 21 - 22

BIEF (7 credits - II sem. - OBBC  |  SECS-S/06)
Course Director:

Classes: 21 (II sem.) - 22 (II sem.)

Class group/s taught in English

Suggested background knowledge

A refresher of differential calculus is suggested.

Mission & Content Summary

An increasing number of economic activities entails financial and probabilistic features that cannot be any more neglected. Several car manufacturers directly supply leases. The leasing cost is summarized in a internal interest rate which represent a sizeable source of the company revenues. Nowadays almost all investment opportunities are accompanied by information on the probability distribution of their yields to maturity. A recent UE legislation states that some accounting items should be determined on the basis of financial and probabilistic principles too. Knowing what a probability and a financial law are, is by now an essential component of the background of every student in Economics. The course objective is to provide students with the basic notions of Probability and Financial Calculus that are required in many Economic, Financial and Management fields. The course consists of three parts: (i) integral calculus – instrumental in facing the second part; (ii) probability calculus – basic notions and their proper use; (iii) financial calculus – basic notions and their application.

  • Integral calculus: antiderivative; indefinite integral; integration methods; definite integral; integral function; generalized integrals and convergence criteria.
  • Probability Calculus: classical, empirical and subjective approaches. Axiomatic approach: sample space, events algebra, probability measure. Conditional probability.
  • Random numbers and vectors: distribution function, probability and probability density functions. Expected value and variance of a random number. Joint and marginal probability function of a random vector; stochastic independence and linear correlation; covariance; expected value and variance of a linear combination of random numbers.
  • Financial calculus: present and final value: financial laws of one and two variables. Decomposability. Annuities and loan amortization. Consumer credit.
  • Fixed income bonds. Interest Rate Term Structure. Duration: financial  immunization and volatility of the bond price.
  • Financial choices: DCF, NPV and IRR. Generalizations: GNPV, APV and GAPV. Financial leverage. Decomposition of global indices.

Intended Learning Outcomes (ILO)
At the end of the course student will be able to...
  • Recognize the proper meaning of standard indices of cost/profitability for a financial operation such as  NPV, IRR,  etc..
  • Identify the proper meaning of probabilistic statements and terms concerning  random quantities such as uncorrelated random yields, default risk and so on.
  • Reproduce the correct procedures  for computing integrals, probabilities and financial quantities.
At the end of the course student will be able to...
  • Apply the learned calculus methods to  compute and/or asses the correctness of quantities which are relevant both in theory and in practice such as: the no arbitrage price of a  bullet bond, the internal effective rate of a loan, the expected returns rate of a portfolio, etc..
  • Evaluate the profitability of a financial operation by choosing the proper method/model to adopt.
  • Compute a probability measure that is coherent with the available information on the stochastic event/number.

Teaching methods
  • Face-to-face lectures
  • Online lectures
  • Exercises (exercises, database, software etc.)

Teaching and learning activities for this course are divided into (1) face-to-face and/or online lectures, (2) in class and/or online exercises (3) self-assessment online materials.  

  1. During the lectures convenient examples and applications allow students to identify the quantitative patterns and their main logical-mathematical properties.
  2. The in class exercises allow students to apply the analytical tools illustrated during the course.
  3. Besides the exercises proposed in class, further exercises, such as "mock exams" and "past written exam" are uploaded online. The online exercises allow students to individually practice and self-assess their own skills.

Assessment methods
  Continuous assessment Partial exams General exam
  • Written individual exam (traditional/online)
  •   x x

    The exam modality is written: the final grade depends exclusively on the student performance in the written exam.

    The written exams consist of both closed-answer and open-answer questions. Their structure is designed in order to assess:

    • The ability to identify the convenient mathematical tool to be used in the described context.
    • The ability to correctly apply the chosen tool to compute the required result.
    • The ability to properly understand the meaning of the learned notions and models 
    • The ability to recognize the main properties and the related logical connections of the main learned notions.


    Every student may take one general exam which is worth the 100% of the final grade. Alternatively students may take two partial written exams (mid-term and end-term exams). In such a case each partial exam is worth 50% of final grade. 



    Teaching materials


    • L. PECCATI, S. SALSA, A. SQUELLATI, Integral Calculus, Extract from Mathematics for Economics and Business, Milano, EGEA, 2008 (Chapter 7).
    • E. CASTAGNOLI, M. CIGOLA, L. PECCATI, Probability. A Brief Introduction, Milano, EGEA, 2009, second edition.
    • E. CASTAGNOLI, M. CIGOLA, L. PECCATI, Financial Calculus with Applications, Milano, EGEA, 2013.


    Further materials will be distributed on Blackboard platform at the beginning of the course.


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