**UNIVERSITY OF ZAGREB**

**DEPARTMENT OF MATHEMATICS**

**Bijenička cesta 30**

10000 Zagreb

**GRADUATE UNIVERSITY PROGRAMME IN FINANCIAL AND BUSINESS MATHEMATICS**

**Zagreb, January 2015**

CONTENTS

1.INTRODUCTION

1.1.Arguments for starting the studies

1.2.Comparability with international programmes

1.3.Experience so far of the Department of Mathematics in conducting mathematics studies

1.4.Studies support the mobility of students

**2.GENERAL INFORMATION**

2.1.Name of the programme

2.2.Performed by

2.3.Duration

2.4.Entry requirements

2.5.Competences acquired at the end of the study

2.6.Learning outcomes at level of the study programme

2.7.Occupations and job posts for which students are qualified

2.8.Undergraduate programmes which are (partially) adequate prerequisites

2.9.Academic title acquired upon completion of courses

**3.PROGRAMME DESCRIPTION**

3.1.The list of courses including the number of classes and ECTS credits

3.2.Course description

3.3.Course structure, rhythm of studying and student's obligations

3.4.List of elective courses from other programmes

3.5.List of courses to be taken in foreign language

3.6.Criteria and conditions for transfer of ECTS credits from other studies

3.7.Completion of studies

3.8.Requirements for continuation of interrupted studies

**4.CONDITIONS FOR EXECUTION OF STUDIES**

4.1.Place of execution of study programme

4.2.Information on premises and equipment foreseen for studies

4.3.Teachers and collaborators to participate in execution of studies

4.4.Information about the teachers participating in the programme

4.5.List of teaching sites for practical teaching

4.6.Optimum number of students considering space, equipment and number of teachers

4.7.Estimated costs of studies per student

4.8.Methods for monitoring quality and performance of study programme execution

## 1.INTRODUCTION

### 1.1.Arguments for starting the studies

Today there is no doubt that mathematics is one of the essential sciences for the development of human civilisation. It was generated from everyday needs of old Egyptian people in 20th Century B.C. (counting of various object, land measuring, construction of dwellings, celestial and phenomena at sea, the beginnings of trade and barter), which lead to deliberation on size relationship and space shapes. Today, mathematics represents “the queen of all sciences”, developing powerfully and rapidly over an enormously wide and complex field of research and application with a considerable number of scientist and experts. Its importance in contemporary society is reflected by the fact that, besides native tongue, mathematics is the most represented school subject in almost all schools all over the world.

Mathematics is traditionally tied with unbreakable bonds with natural sciences, particularly with physics, as well as with technical sciences which have served as source for innumerable examples and inspiration for its own ideas and development. However, recently we have witnessed the ever increasing breaking of mathematics into economics, medicine, psychology, linguistics and other scientific fields. It is impossible to even consider performing serious research in all those fields without various quantitative and analytical, complex combinatorial, probabilistic, statistical and other mathematical methods and techniques as well as mathematical models and simulations. It is sufficient to say that many of Nobel Prize winners for economics have been, by their basic education, mathematicians and that the biggest scientific achievements in modern medicine have been achieved through joint work of interdisciplinary research teams with a considerable contribution of mathematicians. Besides the increasing influence of mathematics, there is rapid development of information and communication technologies where mathematics has been participating from the beginning. It is a well-known fact that mathematicians were principal intellectual initiators for developing the idea of a computer and its construction; therefore, they were the initiators of an “information revolution” which has changed human society radically. Finally, modern ways of communication and information search by computer networks (i.e. Internet) would not be possible without mathematical algorithms developed for that purpose. Because of the scope and diversity of the research and application, instead of mathematics today we more often speak of mathematical science.

We would like to emphasise scientific research in financial mathematics as one of the most striking examples for the application of sophisticated mathematical methods to very concrete practical problems. Large number of researchers and experts in the fields of finance and mathematics has been intensively working for twenty years on various problems, which are important for the analysis of financial security papers and financial and insurance risks. As an example, we would like to mention three multidisciplinary magazines launched in the last fifteen years as the result of the scope of such research: *Mathematical Finance (1991), Finance and Stochastic* (1997) (Springer-Verlag), *Insurance: Mathematics and Economics.*

Modern life all over the world, and in Croatia as well, could not be imagined without mathematics and mathematicians. Today, mathematicians are to be found in all segments of Croatian economy and science. They are employed by information and financial sectors (computer centres, companies dealing with software manufacture for the most diverse purposes, insurance companies, pension and other funds, etc.) and elsewhere. Many of them work at different higher education institutions – universities and polytechnics, since almost all study programmes include mathematics courses. Mathematics is the compulsory subject at all primary and secondary schools; therefore, many mathematicians work in schools.

The consequence of such a wide selection of possible employments, as well as the need for experts in the field of mathematics and information and communication technologies, is that mathematicians have been successfully employed in Croatia. This ranks the Faculty of Science, Department of Mathematics among the rare Croatian higher education institutions which provide its students the possibility of successful employment after graduation. World experience shows that this trend is not accidental and fleeting. On the contrary, the demand for mathematicians of various specialisations in labour market, (both public and private sector), will continue to grow

Therefore, it is unnecessary to additionally motivate the need for university studies of mathematics of all levels (undergraduate, graduate and postgraduate) at the University of Zagreb and for their constant improvement and updating.

The Graduate University Programme in Financial and Business Mathematics qualifies mathematicians for the employment in economic surrounding. With the usual wide knowledge of various mathematical disciplines, which would give them quantitative skills and analytical way of thinking, graduated students shall have basic knowledge in economics as well. The existence of such experts is required by the ever developing Croatian market economy, and it shall particularly contribute to rapid development of sophisticated financial markets.

Proposed studies are oriented towards the application of mathematics in finances and insurance, with special emphasis on stochastics and statistics, optimisation and operations research. At the same time, students take a considerable number of fundamental courses in economics. A part of content to be taught at this graduate studies is directly founded on the results of the most recent research in the field of financial mathematics and shall qualify students to use basic quantitative skills. Since experts with quality educations are still scarce, we estimate that students who graduate in these courses will be able to get employment without any difficulties either in public or private sector. Potential employers are insurance companies, pension and health funds, government offices and agencies, Croatian National Bank, commercial banks, exchange and brokerage houses.

Traditionally, mathematicians qualified in this field most frequently found employment in insurance companies as actuaries. The development of more complex insurance products created additional need for such experts. Similarly, pension and health funds in Croatia represent another employment opportunity for future Masters in Financial and Business Mathematics, particularly in the light of current reforms. Government offices and agencies already have great need for such experts both for the regulatory and supervisory aspect of their activities. Banks also experience the growing need for analytical methods in risk management, financial assets management, analysis of financial instruments etc. Finally, we expect exchanges and broking houses in the region to create job opportunities for our future graduates by adopting more sophisticated analytical methods of managing their risk and securities.

### 1.2.Comparability with international programmes

In May 2001, the European Commission approved the pilot project *Tuning Educational Structures in Europe. *The Project was coordinated by the Universities of Deusto and Groningen and 105 European universities with 128 academic representatives from the fields such as business and management, education, geology, mathematics, chemistry and physics participated in the project. Their aim was to establish the largest possible degree of European unity, i.e. the common platform in university undergraduate and graduate education in these seven fields.

Among others, the international group of experts for harmonization of university mathematics programmes, the so-called The Mathematics Tuning Group, was established. Its guidelines were published in the following document:

The Mathematics Tuning Group: *Towards a common framework for Mathematics degrees in Europe*, E.M.S. Newsletter 45 (2002), pp. 26-28.

Accordingly, our basic principle in shaping the university undergraduate mathematics course was to follow as closely as possible those suggestions hoping that mathematicians from most European university and mathematics courses would follow as well. The comparison of our undergraduate studies with corresponding foreign universities should be considered within that context.

Similar programmes are extremely popular in the world in the last ten to fifteen years, mostly because the attendants can easily find well paid employment. These programmes were first introduced in the United States and then in European Union countries. As an example of analogue programmes in Europe, here are some:

*Finanz- und Wirtschaftsmathematik Diplom,*Technische Universität München, Germany*BSc/Msci Mathematics with Economics,*University College London, Great Britain

Both programmes are conceptually, by their basic principles and course content structure equivalent to the proposed programme

### 1.3.Experience so far of the Department of Mathematics in conducting mathematics studies

Mathematics has continuously been taught at the University of Zagreb for almost 140 years. Presently, more than 1500 students are studying mathematics at the Department of Mathematics at two undergraduate studies, seven graduate and one integrated study. The studies have been organised since 2005. Before 2005 there were two study programmes: *mathematics programme and mathematics and physics programme* (in cooperation with the Department of Physics). Mathematics programme is divided into the following study profiles: *Mathematics Graduated Engineer, Professor of Mathematics and Professor of Mathematics and Informatics*. The first year of studies is common to all stated profiles. Later, at the third year of studies, Mathematics Graduated Engineer profile is divided into the following five directions: *Pure Mathematics, Applied Mathematics, Mathematical Statistics and Computer Science, Computer Science and Financial and Business Mathematics*. The study of mathematics and physics included only one study profile – *professor of mathematics and physics*. All our existing programmes are the successors of the above mentioned studies.

The Graduate University Programme in Financial and Business Mathematics may be viewed as the successor of the former direction *Financial and Business Mathematics within the study profile Mathematics Graduated Engineer* in the third and fourth year of study. In addition, the syllabi have been considerably updated, and teaching and assessment methods have been adjusted to European standards and the Bologna declaration.

### 1.4.Studies support the mobility of students

Graduate University Programme in Financial and Business Mathematicsis open to the mobility of students and teachers. Faculty of Science, Department of Mathematics, makes available all relevant information on teaching plan and programme, study regime and load as well as certificates with achieved results of each individual student, which is an essential prerequisite for transfer of ECTS credits. Details concerning the transfer of ECTS credits from one study programme to another is agreed at the level of ECTS coordinators.

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## 2.GENERAL INFORMATION

### 2.1.Name of the programme

Graduate University Programme in Financial and Business Mathematics

### 2.2.Performed by

Faculty of Science, Department of Mathematics, University of Zagreb

### 2.3.Duration

Two (2) academic years, i.e. four (4) semesters.

### 2.4.Entry requirements

The basic entry requirement for the Graduate University Programme in Computer Science and Mathematics is completion of an adequate university undergraduate programme with at least 180 ECTS credits, which gives necessary competences in mathematics and computer science (for details see Section 2.8). Moreover, classification procedure requirements for enrolment of new students should be fulfilled.

Competences in mathematics and computer science of candidates are assessed by examining curricula of their completed undergraduate studies. Classification procedure involves evaluation of candidate undergraduate grades, as well as special activities (awards, published scientific or professional papers, participation in student competitions in mathematics, physics, computer science or economics, the student assistant status in some undergraduate course, mentoring of high school students taking part in mathematics, physics or informatics competitions, completion of another undergraduate study, two written recommendations given by university lecturers, etc). Students are enrolled according to their total ranking until the quota for that year is fulfilled. Detailed description of classification process for the enrolment of new students in the first year of graduate programme is announced for each academic year at the Department of Mathematics web site and in daily press. Classification procedure does not include any examination.

### 2.5.Competences acquired at the end of the study

The Graduate University Programme in Financial and Business Mathematics, through its course contents as well as its forms and methods of teaching, provides for the acquisition of fundamental knowledge and the understanding of results in the area of financial and actuarial mathematics, probability theory, mathematical statistics, operational research, and optimisation, as well as fundamental knowledge in economics. In addition, through a series of specialised courses, students become acquainted with mathematical methods and models in finance, game theory, decision theory, actuarial science, numerical methods in financial mathematics, monetary economics, financial markets, economics of the public sector, econometrics, etc.

In addition to acquiring theoretical knowledge, students in this programme develop the capacity for independent modelling and creative solving of problems from the economic and financial environment. Students are trained in analytic risk management methods, financial asset management, analysis of financial instruments and insurance products, computation of insurance reserves, and optimal decision-making in a deterministic and stochastic environment.

In this programme, students also develop their ability to understand mathematical proofs, which itself entails the skill of logical argumentation in general situations. The programme also provides a good basis for their further self-education in both mathematics and economics.

### 2.6.Learning outcomes at level of the study programme

Upon successful completion of this programme a student is able to:

I.demonstrate intuitive and formal knowledge and understanding of basic concepts and results in financial and actuarial mathematics, probability, mathematical statistics, operations research, optimization and economics;

II-1. argue mathematically, interpret a mathematical proof, and construct a proof of a new mathematical statement;

II-2.independently model and solve problems in economic and financial environment;

II-3.apply financial, numerical and statistical programme packages in interpreting and analyzing dana;

II-4.present a mathematical content in written and oral form using mathematical language and notation, and contemporary computer technology;

III.independently use mathematical literature in Croatian and English, and deliver a professional project and paper, both individually and in a team;

IV. take responsibility for their own learning, further university education, and professional development.

### 2.7.Occupations and job posts for which students are qualified

Holders of the degree of Master of Mathematics in Financial and Business Mathematics are qualified for employment in insurance companies (as actuaries), banks, stock exchanges and brokerage firms (risk analysis, securities analysis, asset management), pension and health insurance funds, government offices and agencies, but also in the scientific research and higher education system (universities, polytechnics, research institutes) in entrance-level teaching or research positions (asistent/ica, znanstveni novak/novakinja, stručni suradnik/suradnica).

2.8.Undergraduate programmes which are (partially) adequate prerequisites

Prerequisite for quality performance at Graduate University Programme Financial and Business Mathematics is solid mathematical background acquired in undergraduate studies, i.e. good knowledge and understanding of differential and integral calculus, function of one and several variables, linear algebra and main results of basic (classic and modern) fields of mathematics, with the emphasis on differential equations, complex functions, measure theory, probability, statistics and discrete mathematics. Further competencies important for the graduate programme are: the ability to understand proofs, the ability to mathematically model a situation, the ability to solve problems using mathematical tools and the solid background in the use of information technology.

In our opinion the following programmes satisfy these requirements:

#### a.at Department of Mathematics, University of Zagreb:

-**with direct entry**(no differential exams and study modules): the University Undergraduate Programme in Mathematics

-**with possible substitute elective courses:**theUndergraduate University Programme in Mathematics Education. Explanation: students instead of one of the elective courses offered must enter as an elective subject Measure and integration from Undergraduate study of Mathematics with the responsibility of taking the item, in order to bring their knowledge to the level of the intended programme of study.

#### b.at other institutions in Republic of Croatia:

-**with direct entry**(no differential exams and study modules): any university undergraduate programme in Mathematics