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


A.Y. Credits
2016/2017 8
Lecturer Email Office hours for students
Laura Gardini

Assigned to the Degree Course

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

Learning Objectives

The course aims to complete the training provided by the basic course, introducing new financial evaluation techniques for financial flows and government bonds and assets (financial immunization techniques). Some basic elements are introduced for the estimation and evaluations of financial transactions within uncertainty (or risky assets). The knowledge of basic mathematical tools is improved with the introduction of some properties of real functions in several variables, optimization without constraints and with constrains. These properties are applied to solve the problem of optimal portfolio selection of risky assets (of the Markowitz model).


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 acquiredStudents 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
Regarding the topics covered in the course, the advanced financial sector, the student must acquire the knowledge and mathematical tools for the understanding of key financial variables and their application in the models of portfolio theory.
The verification of learning takes place through talks in the classroom and at the examination.

Applying knowledge and understanding
The student must be able to apply the acquired knowledge, even under slightly different contexts, and to understand and solve realistic optimization problems applied to financial mathematics. Examples are shown in the classroom during lessons and exercises.

Making judgements
The student must have the ability to use the acquired knowledge to independently solve optimization problems in the financial context that may appear new. Examples of such applications are shown in the classroom during lessons and exercises.

Communication skills
The student must be able to clearly communicate own claims and considerations of portfolio theory problems. The working mode is shown in the classroom during lessons and during exercises.

Learning skills
The student must have learned the material so that to be autonomous at work or to continue in further advanced studies, having acquired the skills needed to develop new knowledge and professional skills. Examples are provided in the classroom.

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.
Parte B e C: Pantry distributed by the teacher.


The exam consists of a written work, answering to six questions delivered by the teacher on the program.

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