Università degli Studi di Urbino Carlo Bo / Portale Web di Ateneo


A.Y. Credits
2021/2022 8
Lecturer Email Office hours for students
Laura Gardini By appointment to schedule via email.
Teaching in foreign languages
Course with optional materials in a foreign language English
This course is entirely taught in Italian. Study materials can be provided in the foreign language and the final exam can be taken in the foreign language.

Assigned to the Degree Course

Economics and Management (LM-77)
Date Time Classroom / Location
Date Time Classroom / Location

Learning Objectives

The course aims to complete the preparation provided by the basic course, introducing new assessment tools for financial flows and bonds (financial immunization techniques). Some basic elements are the introduced for the estimation and evaluation of financial transactions within uncertainty (or risky assets). The mathematical subjects include some properties of real functions of several variables, and elements of free and constrained optimization. These properties are applied to the problem of selecting an optimal portfolio of risky assets.


Part A (Evaluation without uncertainty)

Elements of bond portfolio management: the Market Consistency Hypothesis (no Arbitrage, short sales not allowed). Term structure, spot rates and forward rates. Lengthening of the structure by maturity and "coupon effect". Evaluation of flows using the term structure. Estimate of the present value volatility. Duration and volatility of the Value function. Its use in financial immunization techniques.

Part B (Static Optimization Tools)

Uncostrained classical optimization, Method of Least Squares, Convex functions, conditions of convexity for quadratic forms, and under affine constraints. Optimization with Equality Constraints. minimum risk portfolio with n risky securities.

Part C (Evaluations under uncertainty)

Evaluation Criteria under uncertainty. Mean Value criterion. Paradox of St. Petersburg. Expected utility criterion. Risk aversion/propension by use of the utility function. Some examples of utility functions. Mean-Variance criterion (M-V). Risk-return analysis. Indifference curves, with examples.

Portfolio Theory under uncertainty of returns. Portfolio of two risky assets (not correlated or correlated). Short sales. Selection of an optimal portfolio. Optimal portfolio with n risky assets. Feasible region and properties of the efficient frontier. Portfolio with one risky asset and one risk-free (Bond). Portfolio with n risky assets and one risk-free (CAPM, Capital Asset Pricing Model). Capital Market Line. Separation theorem. The benefit of diversification.

Learning Achievements (Dublin Descriptors)

Learning outcomes and competences to be acquired

Students are expected to acquire knowledge and understanding in applied mathematics to the theory of portfolio of risky assets and elements useful in the evaluation of flows also under uncertainty and risk, so to get a proper knowledge suitable for autonomy in working environments in the financial sector. In addition to a good learning ability, it is expected the ability to apply the acquired knowledge in autonomous and competent way.

Knowledge and understanding

At the end of the course, the student must have mastered the maths of the advanced level financial sector that is being dealt with in the course. It should be able to know the mathematical tools suitable for understanding the main financial variables and their use in the computation models applied in portfolio theory. Examples and working modes are shown in the classroom during lessons.

Applying knowledge and understanding

At the end of the course the student must have acquired a good ability to use the mathematical and financial tools studied and know how to use them in situations similar to those presented in the course. He/she should be able to properly apply the formulation studied and be able to solve portfolio theory problems similar to those studied. In particular, he/she must be able to apply the acquired knowledge even in contexts slightly different from those studied and have the ability to use acquired knowledge to solve problems that may appear new. Examples of such applications are shown in the classroom during lessons.

Making judgements

At the end of the course, the student must have acquired a good ability to analyze subjects and problems of portfolio flow and theory evaluation, the ability to critically evaluate any proposed solutions, and correct interpretation of similar arguments.

Communication skills

At the end of the course, the student must have acquired a good ability to clearly communicate own considerations regarding optimization and portfolio theory. The working mode is shown in the classroom during lessons and exercises.

Learning skills

At the end of the course the student must have acquired a good degree of autonomy in the study of the discipline, in the reading and interpretation of financial data, in the search for useful information to deepen the knowledge of the discussed topics.

Teaching Material

The teaching material prepared by the lecturer in addition to recommended textbooks (such as for instance slides, lecture notes, exercises, bibliography) and communications from the lecturer specific to the course can be found inside the Moodle platform › blended.uniurb.it

Didactics, Attendance, Course Books and Assessment

Course books

Part A: Ch. 6 and 7 from “R.L. D’ecclesia e L. Gardini, Appunti di matematica finanziaria. Giappichelli, Last Edition, or Pantry distributed by the teacher.

Part B and C: Pantry distributed by the teacher.


The exam consists of a written test, answering 5 open questions, including 2 questions on Part A and at least 2 questions on Part C. The final evaluation is determined by adding the scores obtained by evaluating from 0 to 6 points each answer to the 5 questions. The assessment of the answer is based on the different levels of knowledge of the subject matter: knowledge of the meaning and computation method of the subject matter, knowledge of theoretical arguments and of the proofs leading to the result.

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